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Cross-Section Estimation of Long-Run Relations Using Time-Compressed Data
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\title{Cross-Section Estimation of Long-Run Relations Using Time-Compressed Data}
\author{ Serena Ng\footnote{Dept. of Economics, Columbia University and NBER. 1126D-420 W. 118 St., New York, NY 10025, Email: serena.ng at columbia.edu} \and Nikolay Gospodinov\thanks{Research Department, Federal Reserve Bank of Atlanta, Email: [email removed] \smallskip \newline Financial Support from the National Science Foundation (SES 2018369) is gratefully acknowledged. The views expressed here are the authors' and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System.} }
\date{\today}
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\bibliographystyle{harvard}
\begin{abstract}
Many empirical investigations of long-run relations are based on cross-section regressions in averaged or long differenced data that effectively have the time dimension of a $T\times N$ panel compressed. We analyze a class of time-compressed I(1) data and show that they have magnified variability stemming from the fact that the cross-section variance of a non-stationary panel `fans out' with time. Cross-section regressions in time compressed data can potentially yield estimates that are super-consistent and asymptotically normal, whether the regressors are stationary, non-stationary, or highly persistent. The fastest convergence rate of $\sqrt{N}T$ requires a compression scheme that not only magnifies the non-stationary signal, but also dilutes the regression noise. Omitted fixed effects preclude noise dilution but the estimates remain super-consistent. However, the fanning out effect can be weakened when the data have a strong force for mean-reversion or convergence, a problem that seems relevant for temperature data. We consider three applications and find that the long-run relation between consumption and income, and between growth/inflation and demographic variables are reasonably well determined, but the estimated relation between growth and warming temperature is fragile.
\end{abstract}
\bigskip
\noindent Keywords: long-run multiplier, climate, aging, Ricardian regressions, non-stationarity, long-differencing, fanning-out.
\bigskip
\noindent JEL Classification: C01, O5, Q5,
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\section{Introduction}
Many long-run economic relations are of interest:- between consumption and output, productivity growth and real interest rates, public debt and GDP, purchasing power parity, inflation and interest rate, inflation and demographics, interest rates of different maturities. While it is common to study these relations using time series or panel data estimators with known theoretical properties,\footnote{See, for example, \citet{pesaran-etal:07}, \citet{lunsford:17}, \citet{yi-zhang:17}, \citet{yoon-kim-lee:14}, and \citet{juselius-takats:21}.} cross-section regressions using data averaged over long periods of time have also been employed. Examples include investigation of savings and investment in \citet{feldstein-horioka}, growth and its determinants considered in \citet{feldstein-horioka, barro-91, easterly-levine} and \citet{mendelsohn-etal:94}. Long-differences of averaged data are also increasingly used to estimate `longer run' relation between economic and environmental variables, as in \citet{dell-jones-olken:12, burke-emerick:16}. The data after averaging and differencing used in these regressions essentially have no time dimension, or what we will refer to as {\em time compressed}. What are the properties of the time compressed data? Is one method better than another? How effective are such data in recovering long-run relations? Would non-stationarity matter, and more generally, do time series properties of the data affect cross-section regressions? This paper provides such an analysis.
Our point of departure is the result in
\citet{deaton-paxson-jpe94} that income inequality will increase with age if individual incomes are random walks, with the implication
that the cross-section variance within a cohort will `fan-out' with age. Built on this insight, we study the properties of a class of time compression functions for I(1) (non-stationary) data and obtain three results. First, the cross-section variance of data driven by simple stochastic trends will `fan-out' (or magnified) over time. The maximum fanning out is obtained by `point sampling' the last row of a $T\times N$ panel, while averaging the data over $K$ periods has a shrinkage factor of one-third. Different forms of non-stationarity may change the fanning out profile, but significant variation in cross-section variance can be expected.
Second, the magnified signal from fanning out can be exploited in cross-section regressions. Lemma \ref{lemma:smooth} shows that in the local-to-unity framework, the variance of time compressed data traverses smoothly from $O(1)$ to $O(K)$ as the largest autoregressive root moves towards the unit circle. The convergence rate of the least squares estimator self-adjusts with the degree of persistence.
Even though cross-section regressions do not explicitly use time variations, fast convergence is a direct consequence of the strong signal in highly persistent data. Third, we show that super-consistent (faster than $\sqrt{N}$) and asymptotically normal estimates are usually possible. Proposition \ref{prop:prop1} shows that the fastest convergence rate requires that time compression performs both signal magnification and noise dilution. While K-averaging achieves this rate, point sampling does not because of the inability to dilute noise.
The main appeal of cross-section regressions in time compressed data is that standard normal inference can be used without knowing whether the regressors are I(1) or I(0), with no need for HAC standard errors. In contrast, the asymptotic distributions from time series or panel regressions are either discontinuous if the regressor has a unit root, and/or have nuisance parameters that complicate inference. The cross-section estimates are also robust to endogeneity and low-order misspecification biases because of the magnified signal in the regressors. Cross-section regressions based on time averaged data thus have more desirable theoretical properties than previously understood.
We use three applications to illustrate empirically relevant issues. In the first application that estimates the long-run relation between consumption and income using data for 44 countries from the Penn World Table, omitted fixed effects prohibit noise dilution and preclude the fastest convergence rate, but super-consistent and asymptotically normal estimates can still be attained at the cost of an inflated variance. In the second application, we analyze the relation between GDP growth or inflation taken from the Penn World Table, and demographic variables taken from the World Bank database. The cross-section variance of the dependency ratio `fans-in'. This feature can be traced to convergence across countries of the under-16 population, in spite of clear fanning out of the share of population over 65. In such a case, it is informative to estimate the effect of the two observed components separately. In the third application of growth on temperature using data for the 48 states in the U.S., the trending component is weak relative to the stationary component, and the cross-section variance of temperature shows little fanning out. Unless the two latent components have the same effect on economic outcomes, time compression schemes will be challenged to disentangle the effect of warming trend from regular temperature fluctuations. In this case, the long- and short-run estimates may well appear similar, a finding that has been attributed to a lack of adaptation in the literature. Checking the stability of the estimates with respect to $K$ is effective in gauging whether the long-run relation of interest can be reliably estimated.
The paper proceeds as follows. Section 2 begins with the key insight in \citet{deaton-paxson-jpe94} and Section 3 proceeds to analyze a class of time compression functions that include the important cases of point-sampling and $K$-averaging but exclude long-differencing. Section 4 considers estimation of a static model by cross-section regression in time compressed data under specific assumptions. Section 5 relaxes these assumptions. Section 6 considers mechanisms that can dampen the fanning out in the context of three applications. Section 7 concludes.
\section{Fanning Out of Cross-Section Variations}
This section studies the cross-section
variability in a panel of $N$ non-stationary series observed over $T$ periods. We will let $\mathbb E_i[\cdot]$ and $\text{var}_i[\cdot]$ denote expectation and variance over the cross-section distribution of units, while $\text{var}(\cdot)$ is variance for a given $i$.
Our point of departure is the result in \citet{deaton-paxson-jpe94} that if utility is quadratic and the permanent income hypothesis (PIH) holds, consumption and income inequality will increase with age `$a$'. The result arises because consumption ($c$) is a random walk under PIH, and can be characterized as $c_{ia}=c_{i,a-1}+u_{ia}$. Assuming in addition that $\text{cov}(c_{i,a-1},u_{ia})=0$, the cross-section variance of a given set of $N$ individuals in the same cohort at age $a$ is related to the one at age $a-1$ by $\text{var}_{i|a}(c_{ia})=\text{var}_{i|a-1}(c_{i,a-1})+\sigma^2_i$. It follows
that the cross-section variance of the cohort will increase with $a$, reflecting the cumulative difference in individual shocks as they aged. \citet{ng:08} shows that if a fraction $\kappa$ of the panel is non-stationary, the cross-section variance will still fan out at rate $\kappa t$.
\citet{deaton-paxson-jpe94} noted that other preferences may increase or decrease consumption dispersion over time, and it can even be nullified if individuals are homogeneous.
We now explore the fanning out effect outside of the PIH context, and under more general conditions.
\subsection{Generalized Fanning Out}
Our working assumption is that the data are non-stationary but not necessarily a random walk. Consider
log per-capita real GDP for 44 countries over the sample 1950-2023 taken from the Penn-World Table, \citet{pwt110}. The left panel of Figure \ref{fig:app05_gdp} shows that the time series are non-stationary, and the right panel shows that the cross-section variance fans until 2000 when $\sigma$-convergence began. Many national accounts variables in this database have this property. Ratios such as export-to-GDP, import-to-GDP, debt-to-GDP in the Global Macroeconomic Database provided in \citet{gmd} also have cross-section variances that fan out over a long span.
\begin{figure}[!ht]
\caption{log Real GDP Per-Capita: 44 Countries in PWT}
\label{fig:app05_gdp}
\hspace*{-.75in}
\centering
\includegraphics[width=8.0in,height=2.50in]{do_app05_gdp.png}
\end{figure}
To study the trending cross-section variance, we start with a simple data generating process. For $t=1,\ldots T$ with $x_{i0}=0$, let $\Delta x_{it}=u_{it}$, $u_{it}=C_i(L) \epsilon_{it}$, where $\epsilon_{it}\sim (0,\sigma^2_i)$ is a white noise error with variance $\sigma^2_i$ that is independent of $C_{ij}$ for all $j$, $C_i(L)=1+C_{i1} L+C_{i2}L^2+\ldots$ is a polynomial in the lag operator $L$ satisfying $\sum_{j=0}^\infty C_{ij}^2<\infty$. Using $C_i(L)= C_i(1)+(1-L)C_i^*(L)$, where $C_i(1)=\sum_{j=0}^\infty C_{ij}$ and $C_{ij}^*=-\sum_{k=j+1}^\infty C_{ik}$, the Beveridge-Nelson Decomposition yields
\begin{equation*}
x_{it}=C_i(1)\sum_{s=1}^{t}\epsilon_{is}+C_i^{\ast }(L)\epsilon_{it}=C_i(1)\sum_{s=1}^{t}\epsilon_{is}+w_{it}.
\end{equation*}
In the above,
$x_{it}$ is decomposed into a random walk (non-stationary or I(1)) component $C_i(1)\sum_{j=1}^{t}\epsilon_{ij}$, and a mean-reverting (stationary or I(0)) component $w_{it}=C_i^*(L)u_{it}$.
For a given $i$,
\[
\text{var}(x_{it})= C_i(1)^2 \sigma^2_{i} t + \text{var} (w_{t}),
\]
where $C_i(1)^2\sigma^2_i=\omega^2_i$ is the long-run variance of $\Delta x_{it}$. Thus, unlike an I(0) process when $C_i(1)=0$ and $\text{var}(x_{it})=O(1)$, the variance of an I(1) process increases with calendar time $t$. It follows that the cross-section variance fans-out at rate $t$:
\[ \text{var}_{i}(x_{it})= \mathbb E_{i}[\omega^2_i]\cdot t + \text{var}_i[w_{i,t}]=O(t).\]
Fanning out requires $\mathbb E_i [\omega^2_i]\ne 0$, not that every $\omega^2_i>0$.
Introducing serial correlation in $u_{it}$ changes the pre-factor of the fanning out rate from the average short-run variance $\mathbb E_i[\sigma_i^2]$ to the average long-run variance $\mathbb E_i[\omega^2_i]$. This, however, is not inconsequential. Consider the IMA(1) model $\Delta x_{it}=\epsilon_{it}+\vartheta_i \epsilon_{it-1}$. With a long-run variance of $\omega^2_i=(1+\vartheta_i)^2\sigma_i^2$, the cross-section variance at $t$ is
\[\text{var}_{i}( x_{it})=\mathbb E_i[(1+\vartheta_i)^2\sigma^2_i] \cdot t+\text{var}_i[\vartheta_i^2\sigma^2_i].\]
The closer is $\mathbb E_i[\vartheta_i]$ to -1, the weaker is the fanning out. Two models with IMA(1) as reduced form are: i). the factor model $x_{it}=\lambda_i f_t+v_{it}$, where $f_t=f_{t-1}+u_t$ with $\text{var}_i(x_{it})=\text{var}_i(\lambda_i)f_t^2+\text{var}_i(v_{it})$; and ii). the local level model with $x_{it}=\mu_{it}+v_{it}$, $\mu_{it}=\mu_{it-1}+u_{it}$. In both models, $v_{it}$ is stationary.
\subsection{Convergence and Initial Conditions}
The above simple non-stationary structure may not be a good characterization of some socio-economic data. For example, the cross-section variance in Figure \ref{fig:app05_gdp} stops increasing around year 2000 when emerging and populous economies have grown much faster than matured ones. We now consider alternative forms of non-stationarity and show that some features can interact to generate complex cross-sectional variability.
\begin{itemize}
\item[i] There may be situations when it is appropriate to assume $x_{i,-T_{int}}=0$ instead of $x_{i0}=0$. For example, if $x_{i1}$ is wealth at the start of retirement, then $x_{i,-T_{init}}=0$ when unit $i$ entered the labor force $T_{\text{init}}$ periods earlier. The effective cross-section variance at $t$ is $O(\mathbf t)$, where $\mathbf t=(t+T_{init})$.
\item[ii] Suppose $\Delta x_{it}=g_i+u_{it}$, where $g_i$ is an exogenous drift. From $x_{it}=x_{i0}+g_t t+\sum_{s=1}^t u_{is}$,
\[\text{var}_i(x_{it})=\text{var}_i(x_{i0})+ \mathbb E_i[\omega^2_i] t+ \text{var}_i(g_i) t^2=O(t^2).\]
The faster fanning out rate of $t^2$ arises because the deterministic trend dominates the stochastic trend.
Nonetheless, heterogeneity in $g_i$ is needed to achieve the faster rate.
\item[iii] Suppose that $x_{i0}\ne 0$ and $\text{cov}_i(x_{i0},g_i)< 0$. Then,
\begin{eqnarray*}
\text{var}_i(x_{it})&=&\text{var}_i(x_{i0})+ \bigg(2 \text{cov}_i(x_{i0},g_i)+ \mathbb E_i[\omega^2_i]\bigg) t+ \text{var}_i(g_i) t^2\\
&=& \text{var}_i(x_{i0})+c_1 t + c_2 t^2=O(t^2).
\end{eqnarray*}
If $\mathbb E_i[\omega^2_i]< -2\text{cov}_i(x_{i0},g_i)$, then $c_1<0$ and $\text{var}_i(x_{it})$ may trend down for some values of $t$. But this negative trend can only be temporary because $\text{var}_i(g_i)t^2$ grows with $t$. In such cases, the cross-section variance could exhibit a U-shape with a minimum at $t^*=-\frac{c_1}{2c_2}$.
\item[iv] Suppose $x_{it}=\lambda_if_t+ v_{it}$, $v_{it}$ is stationary, and
$f_t=g_f+ f_{t-1}+u_t$, where $g_f$ is the drift of the common factor $f_t$, and $u_t\sim (0,\sigma^2)$ is stationary and has long-run variance $\omega^2$. Thus,
\[\text{var}_i(x_{it})=\text{var}_i(x_{i0})+\text{var}_i(\lambda_i) f_t^2+\text{var}_i(v_{it})=O(t^2).\]
But $f_t=f_0+g_f t+\sum_{s=1}^t u_s$. If $g_f<0$ and $f_0\ne 0$, $f_t^2=(f_0+g_f t)^2+O_p(t)$ which is quadratic in $t$
will be U-shaped, reaching a minimum at $t^*=-f_0/g_f$ before increasing.
Even with heterogeneity in $\lambda_i$, the cross-section variance may not always increase with $t$.
\end{itemize}
In general, the fanning out profile will depend on data and the sample. There can be episodes when the cross-section variance will not increase, and could even be $O(1)$. To obtain general results, we will assume that $\Delta x_{it}=u_{it}$ where $x_{i0}$ is exogenous, with the understanding that the fanning out rate may be faster or slower than $t$ obtained under alternative assumptions.
\section{A Class of Time Compression Functions}
This section shows that the fanning out effect is a feature of a class of time-compressed data. For each $i=1,\ldots T$ and $t=1,\ldots T$, let $\Delta x_{it}=u_{it}$ with $x_{i0}=0$, and let
\[\tilde X_{iT_1}=\sum_{t=T_0+1}^{T_1} w(t) x_{it} .\]
be defined by a time-compressed function $w(t)$ satisfying the following.
\paragraph{Assumption A:} For $T_0\ge 0$ and $T_1\le T$, (i) $w(t)\ge 0$ for all $t\in[T_0+1,T_1]$; (ii) $\sum_{t=T_0+1}^{T_1} w(t)=1$, and (iii) $w(t)>0$ for some $t\in[T_0+1,T_1]$.
Assumption A.(i) and A.(ii) ensure that $\{w(t)\}$ is a proper probability distribution over the interval $[T_0+1,T_1]$. A.(iii) ensures that the weights are not concentrated in the early sample.
The class of functions include {\em concentrated} weights. An example is point sampling, a one parameter time compression scheme that takes row $T_1$ from a $T\times N$ panel:
\[\tilde X_{iT_1}=x_{i,T_1}=x_{i0}+\sum_{j=1}^{T_1} u_{ij}=O(T_1).\]
Point sampling thus has $w(t)=1_{t=T_1}$, and puts a weight of $\phi_j=1$ on innovation $u_j$ for all $j\le T_1$.
Assumption A.(iii) requires that $T_1$ is not too close to the origin so that enough innovations are accumulated to generate fanning out.
The preceding analysis implies that variability of $\tilde X_{iT_1}$ is maximized by point sampling at $T_1=T$. It is also immediate that $\text{var}(\tilde X_{iT_1})= O(1)$ if $x$ is stationary, so the cross-section variance neither fans in nor out.
Whereas fanning out in point-sampling means that $\text{var}_i(x_{it})$ increases with calendar time $t$, as in \citet{deaton-paxson-jpe94}, fanning-out using other schemes will be taken to mean that $\text{var}_i(\tilde X_{iK})=O(K)$, where $K=T_1-T_0$ depends on at least two choice parameters, $T_0$ and $T_1$.
Such functions include {\em spread-out} weights,
the simplest being full-sample averaging with $T_0=0$, $T_1=T$. But it also includes a three parameter scheme (referred to in the simulations below as DA) that gives a weight of zero to the first $\kappa$ fraction of the sample, and the remaining $1-\kappa$ fraction would receive a weight of $\frac{1}{(1-\kappa) K}$, with $K=T_1-T_0$.
\subsection{A Closer Look at $K$ Averaging}
Many well-cited cross-section regressions are based on time averaged data. For example, \citet{feldstein-horioka} averaged the data over 15 years, while \citet{barro-91} averaged the data over 25 years. The WMO standard for long-run temperature is 30 years.\footnote{See \url{https://wmo.int/media/news/updated-30-year-reference-period-reflects-changing-climate}.} It is thus useful to take a closer look at averaging data over $K$ periods.
We will use $\bar X_{iK}$ to denote time-compressed data by averaging. Averaging observations over time is commonly thought to remove high frequency noise, making it easier to visualize patterns over time. Consider first full-sample averaging where $\bar X_{i,T}=\frac{1}{T}\sum_{t=1}^T x_{it}$. If $x_{it}$ is stationary with long-run variance $\omega^2_i$, then by central limit theorem, $\sqrt{T}\bar X_{i,T}\smash{\mathop{\longrightarrow}\limits^d} N(0,\omega_{i}^2)$. Time averaging of stationary data reduces noise relative to an arbitrary draw $x_{it}$ at any $t$. It follows from $\text{var}(\bar X_{i,T})=O(T^{-1})$ that the cross-section variance $\text{var}_i(\bar X_{iT})=O(T^{-1})$ `fans-in'.
Consider now the I(1) case when $\Delta x_{it}=u_{it}$, where $x_{i0}=0$, $u_{it}\sim (0,\sigma^2_{i})$ is stationary with long-run variance $\omega^2_i$. Then by functional central limit theorem,
$ T^{-1/2} \bar X_{i,T}\Rightarrow \omega^2_i\int_0^1 W(r)dr$, where $W(r)\sim N(0,r)$ is a standard Brownian motion. Since
$E[\int_0^1 W(r)dr]^2=1/3$, for large $T$, we have
$\text{var}(\bar X_{i,T})=\omega^2_i\frac{T}{3}. $
In contrast to the stationary case when $\text{var}_i(\bar X_{iT})$ fans-in, now
\[\text{var}_i(\bar X_{iT})=\mathbb E_i[\omega^2_i]\frac{T}{3}\]
fans-out, but shrinks by 1/3 relative to point sampling at $T$.
To understand the reason for this shrinkage, consider averaging $K=T_1-T_0$ observations between $T_0+1$ and $T_1$, which may be needed if complete data is not available for the full sample, or properties in the sub-sample are of interest. When applied sequentially to a single time series, $K$-averaging is a low-pass filter that passes all variations with periodicities longer than $K$ and attenuates shorter ones.\footnote{ Let $\omega=2\pi/K$. The discrete Fourier transform of the response is $H(w)=\frac{1}{K}\sum_{m=0}^{K-1} e^{-j\omega m}=\frac{1}{K} \frac{1-e^{-j\omega K}}{1-e^{-j\omega}}$. Using $\sin(\theta) = \frac{e^{j\theta}-e^{-j\theta}}{2j}$, the frequency response is the Dirichlet kernel, $H(\omega)\approx \frac{\sin(K\omega/2}{K\omega/2}$ whose first zero is at $\omega=2\pi/K$.} We use $K$-averaging only once for each $i$ to generate
\[\bar X_{iK}= \frac{1}{K}\sum_{t=T_0+1}^{T_1} x_{it}=x_{i,T_0}+ \frac{1}{K}\sum_{t=T_0+1}^{T_1}\sum_{s=T_0+1}^{t} u_{is}.\]
Simplifying the double sum, we obtain
\begin{eqnarray*}
\bar X_{iK}&=&x_{i,T_0}+\sum_{j=1}^{K}\bigg(\frac{K-j+1}{K}\bigg) u_{i,T_0+j}=x_{i,T_0}+\sum_{j=1}^{K} \phi_j u_{i,T_0+j},
\end{eqnarray*}
where $\phi_j=\frac{K-j+1}{K}$ is the weight on innovation $u_{i,T_0+j}$, and $\sum_{j=1}^K \phi_j^2 =\frac{K(K+1)(2K+1)}{6K^2}$. Note that the relevant initial condition for $K$ averaging is $x_{i,T_0}$, not $x_{i0}=0$.
\begin{lemma}
\label{lem:lemma1}
Suppose that $x_{it}=x_{it-1}+u_{it}$ where $u_{it}\sim (0,\sigma^2_{i})$ is independent across $i$ but possibly serially correlated with long-run variance $\omega^2_i$, and $x_{i0}=0$. Let $\bar X_{i,K}$ be the mean of unit $i$ computed from $K$ observations between $T_0$ and $ T_1$. Let $ \bar a_K=\frac{(K+1)(2K+1)}{6K}$. For fixed $K$,
\[ \text{var}(\bar X_{iK})=\text{var}_i(x_{i,T_0})+\bar a_K\omega^2_i,\]
and for large $K$ relative to $T_0$,
$\text{var}_i(\bar X_{iK}) = \mathbb E_i[\omega_i^2] K/3$. \hspace{.25in} $\square$
\end{lemma}
The cross-section variance has two components: one from the variance of the conditional mean, which in this case is the initial condition, and one from the mean of the conditional variance due to the shocks. Now $\text{var}_i(x_{i,T_0})=\mathbb E_i[\omega_i^2] T_0$, and assuming $T_0$ is small relative to $K$ gives $\text{var}_i(\bar X_{iK})\approx \mathbb E_i[\omega^2_i]K/3$ since $\bar a_K/K\approx 1/3$ for large $K$.
\subsection{Properties of $\tilde X_{iK}$}
The results for point sampling and $K$-averaging draw attention to two distinct sets of weights of interest: $\{w(t)\}$ on the data, and $\{\phi_s\}$ on the innovations. This will still be the case as
we now analyze the properties of $\tilde X_{iK}$ for any $w(t)$ satisfying Assumption A.
In the I(1) case, $x_{it}=x_{it-1}+u_{it}= x_{it}=x_{i0}+\sum_{s=T_0+1}^t u_{is}$. Changing the order of summation to sum over $s$ first and then for $t>s$, we obtain
\begin{eqnarray*}
\tilde X_{iK}&=& x_{i,T_0}+\bigg(\sum_{t=T_0+1}^{T_1} w(t)\sum_{s=T_0+1}^t u_{is} \bigg)=x_{i,T_0}+\sum_{s=T_0+1}^{T_1} u_{is}\sum_{t=s}^{T_1} \ w(t)\\ &=&x_{i,T_0}+\sum_{s=T_0+1}^{T_1} \phi_s u_{is}, \quad \quad \text{with} \quad\quad \phi_s=\sum_{t=s}^{T_1} w(t).
\end{eqnarray*}
Note that $ \phi_s$ is the tail sum of $w$ at $s$, and the sequence $\{ \phi_s\}$ is the survival function of the weight distribution $ w$,
starting at $ \phi_{T_0+1}=\sum_{t=T_0+1}^{T_1}w(t)=1$ and decreases to $\phi_{T_1}= w(T_1)\ge 0$. Then,
\[\text{var}(\tilde X_{iK})=\text{var}_i(x_{i,T_0})+\sigma^2_i\sum_{s=T_0+1}^{T_1}\phi_s^2\equiv \sigma^2_i \Phi_K, \quad\quad \Phi_K=\tilde a_K(w) K.\]
Condition A.(iii) ensures that $\sum_{s=T_0+1}^{T_1}\phi_s^2 >0$ for any $K$. In general, $\Phi_K=O(K)$ but the precise magnitude of fanning out depends on $\tilde a_K(w)$ which is scheme specific. Full sample averaging has $\tilde a_K=1/3$, while point sampling has $\tilde a_K=1$.
In the I(0) case, we have, by definition,
\begin{eqnarray*}
\text{var}(\tilde X_{iK})&=& \text{var}_i(x_{i,T_0})+\sum_{s=T_0+1}^{T_1}\sum_{t=T_0+1}^{T_1} w(t) w(s) \text{cov}(x_{it},x_{is}).
\end{eqnarray*}
To make progress, consider the AR(1) case where $x_{it}=\rho x_{i,t-1}+u_{it}=\rho_i^t u_{i0}+\sum_{s=0}^\infty \rho_i^s u_{t-s}$ and $u_{it}\sim (0,\sigma^2_i)$. With $\text{cov}(x_{it},x_{is})=\frac{\sigma^2_i}{1-\rho_i^2} \rho_i^{|t-s|}$ and assuming $\rho_i^t$ is small for large $t$, we have
\[\tilde X_{iK}=\sum_{s=T_0+1}^{T_1} \psi_{s}(\rho_i) u_{is}, \quad\quad \psi_s(\rho_i)=\sum_{t=s}^{T_1} w(t) \rho_i^{t-s}.\]
The function $\psi_s(\rho_i)$ plays the role of $\phi_s$ in the unit root case. We show in the Appendix that
\begin{eqnarray*}
\text{var}(\tilde X_{iK})&=&\frac{\sigma^2_i}{1-\rho_i^2} \sum_{h=-(K-1)}^{(K-1)} \sum_{s,t, t-s=h} w(t) w(s) \rho_i^{|t-s|}\\
&=& \frac{\sigma^2_i}{1-\rho_i^2} \sum_{h=-(K-1)}^{K-1} A_h \rho_i^{|h|}\equiv \frac{\sigma^2_i}{1-\rho_i^2} G_w \equiv \sigma^2_i \Psi_K(\rho_i),
\end{eqnarray*}
where $A_h=\sum_{s,t,t-s=h}^{T_1} w(t) w(s)$ is the total weight placed on observations $s$ and $t$, separated $h$ periods apart.
Under Assumption A, $\sum_{h=-(K-1)}^{K-1} A_h =(\sum_{t=T_0+1}^{T_1} w_t)^2=1$, implying that $\{A_h\}$ forms a probability distribution over periods separated by $h$. Thus,
$G_w(\rho_i)=\mathbb E_{A}|\rho_i^{|h|} \in [0,1]$ is bounded for any $\rho_i\in[0,1]$ and any $w(t)$.
But $\text{var}(\tilde X_{iK})$ increases with $K$ when $x$ is I(1). To transition from a fixed $\rho_i<1$ to $\rho_i=1$, it must be that $\Psi_K=\frac{G_w}{1-\rho_i^2}$ `blows-up' with $K$ via $\frac{1}{1-\rho_i^2}=\frac{1}{(1-\rho_i)(1+\rho_i)}$ as $\rho_i$ approaches the unit circle. In the local-to-unity case when $\rho_i=1-\frac{c_i}{K}$, $G_w=\mathbb E_A[\rho_i^{|h|}]\approx 1-\frac{c_i}{K}\mathbb E_A[|h|]=O(1)$ for any choice of $w(t)$ satisfying Assumption A and any finite $h$. It is
$ \frac{1}{1-\rho_i} \propto \frac{K}{c_i}=O(K) $ that
allows $\Psi_K$ to diverge with $K$ as $\rho_i$ moves towards the unit circle.
\begin{lemma}
\label{lemma:smooth}
Let $\tilde X_{iK}=\sum_{t=T_0+1}^{T_1} w(t) x_{it}$, where $w(t)$ satisfies Assumption A. For given $K$, it holds that
\begin{itemize}
\item[a.] $\psi_s(\rho_i) \;\xrightarrow{\rho_i\to 0}\; w(s)$
and $ \psi_s(\rho_i) \;\xrightarrow{\rho_i\to 1}\; \phi_s$;
\item[b.] $\Psi_K(\rho_i)\rightarrow \Phi_K=O(K)$ as $\rho\rightarrow 1$.
\end{itemize}
\end{lemma}
Part (a) says that the tail sum $\phi_s$ interpolates continuously
between the two extremes.
Part (b) indicates that $\Psi_K$ moves towards the unbounded $\Phi_K$ as $\rho_i$ moves towards the unit circle. Indeed, if $\rho_i=1-\frac{c_i}{K^d}$, then $\frac{1}{1-\rho_i}\propto \frac{K^d}{c_i}$ for any $d>1$.
This allows $\text{var}(\bar X_i)$ to transition smoothly from being bounded to divergent as persistence increases. This result is important to preserve continuity of the asymptotic distribution in regressions that use $\tilde X_{iK}$ as regressor.
\subsection{Long Differencing}
We use $\breve X_{iK}$ to denote data transformed by long differencing (LD) constructed as in \citet{burke-hsiang-miguel:15}, \citet{dell-jones-olken:12}. Precisely, LD is based on the difference of two local averages of $M=2m+1$ observations, one centered at $T_a$, and one at $T_b$:
\[ \breve X_{iK}=\bar X_{iM}(T_b)-\bar X_{iM}(T_a).\]
It thus requires three parameters, $m$, $T_a$ and $T_b$. Though quite widely used, the properties of LD data are not well understood.\footnote{\citet{burke-emerick:16} simulate $x_{it}$ as having a unit-specific deterministic trend and show that if $K$ is sufficiently large, the LD estimates from a cross-section regression are more precise than the ones obtained from a $T\times N$ panel of raw data.
} If $m=0$, the LD(0,$T_a,T_b$) becomes a simple long difference, $\breve X_{iK}= x_{i,T_b}-x_{i,T_a}=\sum_{s=T_a}^{T_b} u_{is}$. Furthermore, if $x_{i0}=0$, LD($0,1,T_b)$ is the same as a point sampling at $T_1=T_b$ and is a member of our time-compression class. With a unit weight on all innovations between 1 and $T$, $\text{var}_i(\tilde X_{iK})=O(K)$. \citet{cochrane-jpe:88} used the variance of a long-difference relative to a short difference to assess the relative importance of the common component. For other parameterizations, the LD weights on the innovations are not so transparent. An example helps. Suppose $T_a=4, T_b=10$, and $m=1$. The observation and innovation weights are:
\begin{center}
\begin{tabular}{lrrrrrrrrrrrr}
\hline
& & 3 & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ & $11$ & $12$\\
\hline
$\breve w(t)$ & $\cdot\cdot$ & $-\frac{1}{3}$ & $-\frac{1}{3}$ & $-\frac{1}{3}$ & $0$ & 0 & 0 & $\frac{1}{3}$ & $\frac{1}{3}$ & $\frac{1}{3}$ & 0 & $\cdot\cdot$\\
$\breve\phi_s$ & $\cdot\cdot $ & 0 & $\frac{1}{3}$ & $\frac{2}{3}$ & $1$ & $1$ & $1$ & $1$ & $\frac{2}{3}$ &
$\frac{1}{3}$ & 0 &$\cdot\cdot$ \\
\hline
\end{tabular}
\end{center}
We see that LD fails Assumption A because $\breve w(t)$ is not always positive and $\sum_t \breve w(t)=0$, not 1. Furthermore, the innovation weights $\{\breve \phi_s\}$ are not smooth; it jumps from 0 to $1/M$ at $T_a-m+1$, and from $1/M$ to 0 at $T_b+m+1$. Perhaps the most notable feature of LD is that it puts a weight of 1 on the
$K-M+1$ innovations that are in the middle, while the observations at the two ends have smaller weights. This `tent' pattern of $\tilde\phi_s$ generalizes, and it can be shown that for $m>0$,
\begin{equation}
\label{eq:eqLDV} \text{var}(\breve X_{iM})= \sigma^2_i\bigg((K-M+1) + 2\sum_{j=1}^{2m} (j/M)^2\bigg)= \sigma^2_i\bigg(K-\frac{(M^2-1)}{3M}\bigg)\approx \sigma^2_i(K-M/3).
\end{equation}
Local averaging of $M$ observations does not change the fact that the fanning out rate is determined by cross-section heterogeneity over $K$ periods. In this regard, LD isolates the same variations as point sampling $K$-averaging. However, when $M$ is large, the effective rate of $K-M/3$ can be much slower than $K$.
The LD combines local-averaging (a low pass filter with zero at $\omega=\frac{2\pi}{M}$, known to attenuate high frequency variations) with differencing (a high-pass filter with zero at $\omega=0$, known to attenuate low frequency variations).\footnote{The squared gain
$ H_{LD}(\omega)|^2\propto \bigg( \frac{\sin (M\omega/2)}{M\sin(\omega/2)}\bigg)^2 4\sin^2 (\frac{\omega K}{2})$ is a product of a function increasing from 0 at $\omega=2\pi/K$ and another function that decreases to zero at $\omega=2\pi/M$. The peak, $\omega^*$, thus lies between the two zones.} It has a peak periodicity of roughly $p^*=\frac{2\pi}{\omega^*}=\frac{2MK}{M+K}$, which is the mean of the two boundary frequencies of $\frac{2\pi}{K}$ and $\frac{2\pi}{M}$. The choice of both $M$ and $K$ are important.
However, the use of 'medium' and 'longer' terminology in the literature is loose. \citet{dell-jones-olken:12} report `medium-term' effects using $m=7$ and $K=15$ which implies $p^*=15$. In contrast, \citet{burke-emerick:16} uses $m=2$ and $K=20$, which implies a lower periodicity of $p^*=8$ in spite of the `longer-term' terminology.
\subsection{Numerical Examples}
We illustrate the extent of signal magnification by simulating data from $x_{it}=\mu_{it}+v_{it}+\theta v_{it-1}$ where $v_{it}$ is stationary. Two models are considered:
\begin{itemize}
\item[1.] local level model, where $\mu_{it}=\mu_{i,t-1}+u_{it}$ with $u_{it}\sim(0,\sigma^2_{u,i})$ and
$q_i=\frac{\sigma^2_{u_i}}{\sigma^2_v}$;
\item[2] factor model: $\mu_{it}=\lambda_{i} f_{t}$, where $f_t=f_{t-1}+u_t$, $u_t\sim(0,\sigma^2_u)$, $\lambda_{i}\sim (0,\sigma^2_\lambda)$ with $q=\frac{\sigma^2_\lambda \sigma^2_u}{\sigma^2_v}$.
\end{itemize}
In both models, shocks to $\mu_{it}$ are uncorrelated with $v_{it}\sim(0,\sigma^2_v)$. The local-level model provides a permanent-transitory decomposition of the data. Since $q_i=\frac{\sigma^2_{u_i}}{\sigma^2_v}$, signal magnification requires the average variability of $\mu_{it}$ to dominate those of the mean-reverting component $w_{it}$.
The factor model provides a common-idiosyncratic representation of the data. The non-stationarity common trend $\mu_t$ is not enough for signal magnification; it also requires the loadings to be sufficiently heterogeneous.
Both have an IMA(1) model $\Delta x_{it}=u_{it}+\vartheta_i u_{it-1}$ as reduced form, where $ \vartheta_i=[-(q_i+2)\pm\sqrt{q_i^2+4q_i}]/2$ is closer to -1 the smaller the signal-noise ratio $q$ is.
\begin{table}[!ht]
\caption{Cross-Section Variance, $\text{var}_i(\tilde X_{i,K})$: $(T_0,T)=(0,50)$}
\label{tbl:csvar}
\begin{center}
Local-Level Model:
\begin{adjustbox}{width=\textwidth}
\begin{tabular}{ll|rrr|rrr|rrr|rrrr} \hline
& & \multicolumn{3}{c}{PS:$\text{var}_i(x_{i,T_1})$} &
\multicolumn{3}{c}{KA($T_0,T_1)$} &
\multicolumn{3}{c}{DA($T_0,T_1$)} &
\multicolumn{3}{c}{LD($m,T_a,T_b$)} \\ \hline
$\sigma_u$ & $\vartheta$& $T_1=10$ & $T_1=30$ & $T_1=T$ & $T_1=10$ & $T_1=30$ & $T_1=T$ & $T_1=10$ & $T_1=30$ & $T_1=T$ & $(0,15,40)$ & $(2,15,40)$ & $(7,15,40)$ \\ \hline
\input table1_locallevel.tex
\hline
\end{tabular}
\end{adjustbox}
\bigskip
Factor Model
\begin{adjustbox}{width=\textwidth}
\begin{tabular}{ll|rrr|rrr|rrr|rrrr} \hline
$\sigma_\lambda$& $\vartheta$ & $T_1=10$ & $T_1=30$ & $T_1=T$ & $T_1=10$ & $T_1=30$ & $T_1=T$ & $T_1=10$ & $T_1=30$ & $T_1=T$ & $(0,15,40)$ & $(2,15,40)$ & $(7,15,40)$\\ \hline
\input table1_factormodel.tex
\hline
\end{tabular}
\end{adjustbox}
\end{center}
{\footnotesize Note: (i) PS is point sampling; (ii) KA is K-averaging with $w(t)=1/K$ for $t\in[T_0+1,T_1]$; (iii) DA averaging uses $w(t)=0$ for $t\in[T_0+1,T_1-m]$ and $w(t)=1/m$ for $t\in[T_1-m+1,T_1]$, $m=K/2$; (iv) LD($m,T_a,T_b)$ is long-differencing of two $m$-period local averages centered at $(T_a,T_b)$. Throughout, $T_0=0$.}
\end{table}
Table \ref{tbl:csvar} reports the cross-section variance of four time compression schemes: (i) point sampling (PS), (ii) $K$-averaging (KA), (iii) an averaging scheme (DA) with $w_t=0$ for $t\in[T_0+1,T_1-m]$ and $w_t = 1/m$ for $t\in[T_1-m+1,T_1]$ where $m=K/2$, $K=T_1-T_0$, and (iv) long-differencing (LD). We evaluate LD$(m, T_a,T_b)$ at $T_a=m$ and $T_b=T_1$ to accommodate the $m$ observations for the centered-moving average so that it is comparable to PS at $T_1$. The results in the first panel confirms that point sampling has more fanning out compared to KA and DA, the two compression schemes satisfying Assumption A. The second panel for $K$ sampling is in line with Lemma \ref{lem:lemma1} that the fanning out is about 1/3 of point sampling. The third panel shows that by reducing the weights on the early observations, the fanning out effect of DA is larger than $K$-averaging that gives equal weight to all observations. The fourth panel shows that fanning out of LD is larger tha K-averaging, but is smaller the larger is $M$. The results also show that the signal magnification effect is smaller the closer $\vartheta$ is to -1.
\section{ Cross-Section Regressions in Time Compressed Data}
This section studies cross-section regression in time compressed data. Let $z_{it}=(x_{it},y_{it})^\prime$ be a vector of $n$ variables characterized by $B_0 z_{it} =B_1 z_{i,t-1}+\epsilon_{it}$, where $B_0$ and $B_1$ are the same across units. For example, when $n=2$,
\[ B_0=\begin{pmatrix} 1 & -\kappa_0 \\ -\beta_0 & 1 \end{pmatrix}, \quad B_1=\begin{pmatrix} \rho & \kappa_1 \\ \beta_1 & \alpha \end{pmatrix}, \quad \epsilon_{it}=\begin{pmatrix} u_{it} \\ e_{it}\end{pmatrix}.
\]
We start with the simple case when $\kappa_0=\kappa_1=\alpha$=$\beta_1$=0 so that letting $\beta_0=\beta$,
\begin{align}
x_{it}&= \rho x_{it-1}+u_{it}\quad &\label{eq:eqx} \\
y_{it}&=\beta x_{it}+e_{it} ,\quad &
\label{eq:eqy}
\end{align}
where $u_{it}\sim (0,\sigma^2_i)$ and $e_{it}\sim (0,\tau^2_i)$ are stationary and uncorrelated. When $\rho<1$, $x$ is stationary (or I(0)), and when $\rho=1$, $x$ is I(1).
If $x_{it}$ is I(1) and $e_{it}$ is stationary, $(1,-\beta)$ is a cointegrating vector in the sense of \citet{engle-granger-87}. A time series regression using $T$ observations of $y$ and $x$ of a particular unit $i$ will converge at the fast rate of $T$, but the asymptotic distribution of $\hat\beta$ is, in general, non-standard. If we focus on the low frequency variations as in \citet{mueller-watson:17}, we have a small sample problem because few cycles with long periodicity will be observed over a short span. Panel regressions also have a fast convergence rate of $\sqrt{N}T$, and estimation can be quite involved if there is cross-section dependence. Instead, we consider least-squares estimation of a `timeless', cross-section between-group regression in time compressed data $\tilde Y_{iK}$ and $\tilde X_{iK}$:
\begin{equation}
\tilde Y_{iK} =c+ \beta \tilde X_{iK} + \tilde e_{ik}.
\label{eq:S}
\end{equation}
We first consider the properties of $\hat\beta$ under the following assumptions.
\paragraph{Assumption B:} (i) $u_{is}\perp e_{it}$ for all $s,t$ and for all $i$; (ii) $u_{it}\sim (0,\sigma^2_i)$ is serially uncorrelated with $\mathbb E_i[\sigma^2_i]>0$; (iii) $e_{it}\sim (0,\tau_i^2)$ is stationary; (iv) $x_{i0}=0$ for all $i$; (v) $\{x_{it},e_{it}\}$ and $(\bar X_{iK}, \bar e_{iK})$ are i.i.d. across $i$ with finite fourth moments.
\paragraph{Assumption C: }(i) $\Phi=\lim_{K\rightarrow\infty} \Phi_K/K=\tilde a_K>0$ and (ii) $A_0=\sum_{t=T_0+1}^{T_1} w_t^2=O(K^{-1})$.
\medskip
We assume that $u_{is}$ and $e_{it}$ are independent across $i$, and for all $s,t$ so that strict exogeneity holds. We do not require $\sigma^2_i>0$ for every $i$, only that $\mathbb E_i[\sigma^2_i]>0$. The condition is needed for identification as it ensures $Q_{XX}=\text{var}_i(\tilde X_{iK})$ is non-degenerate.
Assumption B.(iii) allows $e_{it}$ to be serially correlated but cannot be non-stationary to rule out spurious regressions. Cross-section dependence is not allowed for now and Assumption B.(iv) simplifies the analysis. Assumptions B.(v) allows us to use LLN and CLT for iid variables. In particular, $\frac{1}{N}\sum_i f(\bar X_{iK},\bar e_{iK})\smash{\mathop{\longrightarrow}\limits^{p}} \mathbb E_i[f(\bar X_{iK},\bar e_{iK})]$, giving
\[ \frac{1}{N}\sum_{i=1}^N (\tilde X_{iK}-\bar {\tilde X}_{K})^2 \smash{\mathop{\longrightarrow}\limits^{p}} \text{var}_i(\tilde X_{iK})=\mathbb E_i[\sigma^2_i]\Phi_K\equiv Q_{XX},\]
and
\[\frac{1}{\sqrt{N}} \sum_i (\tilde X_{iK}-\bar {\tilde X}_K)\tilde e_{iK}\smash{\mathop{\longrightarrow}\limits^d} N(0,V_K), \quad \text{where } \quad V_K=\mathbb E_i[(\tilde X_{iK}-\mu_X)^2\tilde e_{iK}^2].\]
The properties of the least squares estimation is determined by
\begin{eqnarray*}
(\hat\beta-\beta) &=&\frac{ N^{-1}\sum_{i=1}^N (\tilde X_{iK}-\bar{\tilde X}_K) \tilde e_{iK}} {N^{-1}\sum_i (\tilde X_{iK}-\bar {\tilde X}_K)^2}.
\end{eqnarray*}
Under Assumption B, strict exogeneity ensures that the estimator is $\sqrt{N}$ consistent when $x$ is I(0) as the denominator of $\hat\beta-\beta$ converges to $Q_{XX}$ which is $O(1)$, and the numerator scaled by $\sqrt{N}$ obeys a central limit theorem. But faster convergence rates can be obtained, and Assumption C makes these requirements precise. When $x$ is I(1), we have a {\em signal magnification} effect that is scheme specific if C.(i) is satisfied since $\text{var}(\tilde X_{iK})=\sigma^2_i \tilde a_K(w)K$. Now $\text{var}(\tilde e_{iK}) =\text{var}(\sum_t w(t) e_{it})=\tau_i^2\sum_t w(t)^2= \tau_i^2 A_0(w)$. If the time compression scheme is such that $\text{var}(\tilde e_{iK})=O(K^{-1})$ as indicated in C.(ii), we have {\em noise dilution}. Obviously, if we point sample at $t=T_1+1$ and $T_1$ is small, then $\Phi_K=1$ and both C.(i) and C.(ii) would fail.
If we point sample at a large $T_1$, C.(i) is satisfied, but C.(ii) fails because are concentrated to one point, and $\sum_t w(t)^2=1$. Time compression schemes with spread out weights satisfy both C.(i) and C.(ii), and of these, $K$-averaging is the simplest.\footnote{To check C.(ii) in \textsc{matlab}, compute \texttt{sum(w.\textasciicircum 2)} which should be $O(1/K)$. To check C.(i), compute \texttt{sum(cumsum(w.\textasciicircum 2))} which should be $O(K)$.}
\begin{proposition}
\label{prop:prop1}
Let $\hat\beta$ be estimated from a correctly specified static regression $\tilde Y_{iK}=c+\beta \tilde X_{iK}+\tilde e_{iK}$ where $(\tilde X_{iK},\tilde Y_{iK})=(\sum_{t=T_0+1}^{T_1} w(t) x_{it}$, $\tilde Y_{iK}=\sum_{t=T_0+1}^{T_1} w(t) y_{it})$. Assume $\text{var}(x_{i,T_0})=o(1)$.
\begin{itemize}
\item[a.] Let Assumptions A and B hold. As $N\rightarrow\infty$ with $K$ fixed and whether $x$ is I(1) or I(0), (i)
$\hat\beta\smash{\mathop{\longrightarrow}\limits^{p}} \beta$ and (ii) $\sqrt{N}(\hat\beta-\beta)\smash{\mathop{\longrightarrow}\limits^d} N(0, Q_{XX}^{-1} V Q_{X}^{-1})$, where $V=\text{var}_i((\tilde X_{iK}-\mu_X)\tilde e_{iK})$.
\item[b] Let Assumption A, B and C.(i) hold and $x$ is I(1). Let $(\tilde X_{iK},\tilde Y_{iK})$. For large $K$,
$\sqrt{NK}(\hat\beta-\beta) \smash{\mathop{\longrightarrow}\limits^d} N(0,\mathcal Q_{XX}^{-1} \mathcal V \mathcal Q_{XX}^{-1})$, $\mathcal V=\lim_{K\rightarrow\infty} V$, $\mathcal Q_{XX}=\lim_{K\rightarrow\infty} K^{-1} Q_{XX}$.
\item[c] Let Assumption A, B, and C.(ii) hold and $x$ is I(1). Then for large $K$, \newline
$\sqrt{NK}(\hat\beta-\beta)\smash{\mathop{\longrightarrow}\limits^d} N(0,\mathcal Q_{XX}^{-1} \mathcal V_1 \mathcal Q_{XX}^{-1})$, $\mathcal Q_{XX}=\lim_{K\rightarrow\infty} Q_{XX}$ and $\mathcal V_1= \lim_{K\rightarrow\infty} K^{-1} V$.
\item[d] Let A, B, and C hold and $x$ is I(1). Let $(\tilde X_{iK},\tilde Y_{iK})=(\bar X_{iK},\bar Y_{iK})$ under K-averaging.
For large $K$, $\sqrt{N}K(\hat\beta-\beta) \smash{\mathop{\longrightarrow}\limits^d} N(0,\mathcal Q_{XX}^{-1} \mathcal V \mathcal Q_{XX}^{-1})$, $\mathcal V=\lim_{K\rightarrow\infty} V$, $\mathcal Q= \lim_{K\rightarrow\infty} K^{-1} Q_{XX}$.
\end{itemize}
\end{proposition}
The statement on $\text{var}(x_{i,T_0})$ makes clear that limiting variation is due to a large $K$ and not a large $T_{init}$.
As stated in part (a), $\hat\beta$ is $\sqrt{N}$ consistent and asymptotically normal for fixed $K$ regardless of whether $x$ is I(1) or I(0). Consistency follows from the fact that $u_{it}\perp e_{is}$ by assumption, and $\text{cov}_i(\tilde X_{iK},\tilde e_{iK})=0$ for an $K$. If $K=1$, $K$-averaging and point sampling reduce to the standard cross-section result that $\hat\beta$ is $\sqrt{N}$ consistent and asymptotically normal, even when $x$ is I(1).
Recall that $\text{var}(\tilde e_{iK})= \tau_i^2 A_0(w)$ and $\mathbb E_i[(X_i-\mu_X)^2]=Q_{XX}=\mathbb E_i[\sigma^2_i]\Psi_K$.
Under homoskedasticity, the asymptotic variance of $\hat\beta$ is simply $\frac{\tau^2}{\sigma^2} \frac{A_0(w)}{ \Psi_K}$ when $x$ is I(0). The quantity $\frac{\tau^2 }{\sigma^2}$ is the noise in the model relative to the signal in the data. But from Lemma \ref{lemma:smooth}, $\Psi_K\rightarrow \Phi_K=\tilde a_K(w) K$ as $\rho\rightarrow 1$. Thus, when $x$ is I(1) or is highly persistent,
\begin{equation}
\label{eq:eqVhomo}
\sqrt{N}(\hat\beta-\beta) \smash{\mathop{\longrightarrow}\limits^d} N\bigg(0,\frac{\tau^2}{\sigma^2} \frac{A_0(w)}{ \tilde a_K(w) K}\bigg).
\end{equation}
Because the cross-section variance fans out, a faster convergence rate is now possible when $K$ is large. The precise rate depends on how fast the variance of $\hat\beta$ tends to zero and is scheme specific through $w(t)$.
If there is signal magnification but no noise dilution as in point sampling at large $K=T_1$, then as indicated in part (b), $\hat\beta$ is $\sqrt{NK}$ consistent. It is possible for $x$ to be I(1) and yet there is no signal magnification. As indicated in part (c), the estimator is $\sqrt{NK}$ consistent provided there is noise dilution. If both conditions of Assumption C are satisfied, the estimator is $\sqrt{N}K$ consistent. The fastest convergence rate obtains when $K=T$.
Proposition \ref{prop:prop1} assumes that the cross-section variance of $x$ fans out at a rate of $K$, but the rate can be slower or faster under alternative assumptions, as discussed above.
In the case of an exogenous drift $g_i$ for example, $\bar X_{iK}-\mu_X= \frac{K}{2}(g_i-\mathbb E_i(g))+\sum_{s=T_0+1}^t u_{is}=O(K)$, $\bar e_{iK}=O(K^{-1/2})$. The meat of the sandwich variance is
$V_K=O(K)$, and the asymptotic sandwich variance is $O(K^{-4})O(K)=O(K^{-3})$. It follows that $\hat\beta$ from the static regression in K-averaged data is $\sqrt{N}K^{3/2}$ consistent and asymptotically normal, faster than the $\sqrt{N}K$ rate when $g_i=0$.\footnote{The result parallels the convergence rate of $T^{3/2}$ in a time series regression of $x$ on a linear trend. However, if $\text{var}_i(g_i)=0$, the only source of fanning out is the stochastic trend and the convergence rate remains $\sqrt{N}K$.
If $\mathbb T_1>K$, $\text{var}(\bar X_{iK}) =\mathbb T_1 \mathbb E_i[\sigma^2_i]$.}
Another example is when $x_{i,T_{init}}=0$, then with $\mathbb T_1=T_{init}+T_1$, point sampling gives $Q_{XX}=\mathbb T_1 \mathbb E_i[\sigma^2_i]$ and $ V_K\approx \frac{\mathbb T_1}{K} \mathbb E_i[\sigma^2_i\tau_i^2].$
Thus, $Q_{XX}^{-1} V_K Q_{XX}^{-1}=O(\mathbb T_1^{-1})$, giving $\sqrt{N\mathbb T_1}$ consistent and asymptotically normal estimator as $\mathbb T_1\rightarrow\infty$ with $K$ fixed. When the fanning out rate is slower, the fastest convergence will not be attainable, but it cannot be slower than $\sqrt{N}$ provided that $\text{cov}_i(\tilde X_{iK},\tilde e_{iK})$ is of lower order than that of $\text{var}_i(\tilde X_{iK})$.
In spite of the possibility of many convergence rates, knowledge of the rate is not needed for inference, nor is there a need for HAC standard errors. The reason is that the scores $(\tilde X_{iK}-\mu_X)\tilde e_{iK}$ are iid across $i$. Hence a CLT for iid data applies in spite of serial correlation structure of $e_{it}$, which is already absorbed in $V$. Asymptotic normality implies that the usual $t$ statistic can be used for inference. It would be a simple Wald test to see if the estimates in non-overlapping samples are equal. Nonetheless, Assumption B.(iii) assumes $u_{it}\sim (0,\sigma^2)$ is serially uncorrelated. With serial correlation, $\sigma^2_i$ would be replaced by $\omega^2_i$. In the IMA(1) case, $\omega^2_i=(1+\vartheta_i)^2\sigma_i^2$. If $\vartheta_i=-1+\frac{\bar\vartheta_i}{\sqrt{T}}$ as in the `nearly integrated, nearly white noise' case considered in \citet{perron-ng-restud}, then $\omega_i^2 \tilde a_K(w)K = \bar \vartheta_i^2 \tilde a_K$. With $Q_{XX}=\mathbb E_i[\omega^2_i] \tilde a_K K=\mathbb E_i[\bar\vartheta_i^2]\tilde a_K$, there will be no fanning out at all, and a $\sqrt{N}$ consistent estimate is all that can be achieved if there is no noise dilution. However, with noise dilution, a $\sqrt{NK}$ rate can be attained as indicated in part (c) of Proposition \ref{prop:prop1}.
The LD is not in the class of functions considered, but some observations can be made. The LD is similar to the filter used in \citet{kuznets:58} to suggest evidence of a 20-year economic cycle. \citet{howrey:68} noted that the ten-year difference of two five-year moving-averages could spuriously induce a cycle with a 20 year periodicity even if the raw data were white noise. Use of long-differenced data in time series regressions is also known to suffer from small sample biases, \citet{campbell-jef:01}. However, the LD transformation is used only once at $t=T_a$ and $T_b$ to collapse the time variations into $\breve X_{iK}$. The LD regression error is $\breve e_{iK}= \bar e_{iM}(T_b)- \bar e_{iM}(T_a)$. While noise is diluted at rate $M$, $\text{var}_i(\breve e_{iK})=2\tau_i^2/M$ is inflated because of differencing. Since $\text{var}(\breve X_{iK})\approx \sigma^2_i (K-M/3)$ for LD, it has sandwich variance $\frac{2\tau^2}{M} Q_{XX}^{-1}$ under homoskedasticity. Compared to $K$-averaging which has $\text{var}_i(\bar e_{iK})=\tau_i^2/K$ and $\text{var}(\bar X_{iK})=\sigma^2_i K/3$,
\begin{eqnarray*}
\frac{\text{var}(\text{LD})}{\text{var}(\text{KA})}= \frac{2K(T_0+a_K)}{ M(K-M/3)}\approx \frac{2(3T_0+K)}{3M(1-M/(3K))}\approx\frac{2K}{3M}+\frac{2T_0}{M}.
\end{eqnarray*}
This ratio only depends on the tuning parameters and not on $\sigma^2_i$ or $\tau^2_i$. K-averaging should have a smaller variance when $\frac{K}{M}>\frac{3}{2}$.
\begin{figure}[ht]
\caption{Density of $\frac{\hat\beta-\beta}{\hat\sigma_{\hat\beta}}$}
\label{fig:thm1}
\begin{center}
\includegraphics[width=6.5in,height=2.0in]{do_figure_thm1.png}
\end{center}
\end{figure}
To verify the theoretical predictions, we simulate data as
\begin{eqnarray*}
y_{it}&=&\beta x_{it} + e_{it}\\
x_{it}&=& x_{it-1}+v_{it}+\vartheta v_{i,t-1},
\end{eqnarray*}
where $(e_{it}, v_{it})$ are mutually uncorrelated Gaussian random variables with $(\sigma_e,\sigma_v)=( 5, 0.5)$. We consider full sample averaging and point sampling at $T$. To make maximum use of the data with $(T,N)=(50,50)$, we use $T_b=T-m-1, T_a=m+1$ for LD.
Figure \ref{fig:thm1} plots the density of $ (\hat\beta-\beta_0)$ standardized by White's standard errors for the case $\vartheta=0$ and $\beta_0=-1$. The PS (in red), KA (in blue), and LD (in green) are all well approximated by the standard normal distribution. In spite of the similarity in the normalized distribution, there are distinctive differences in standard errors, being 0.014, 0.001, and 0.006 for PS, KA, and LD, respectively. The convergence rate of each estimator is determined by the rate at which its variance tends to zero. For this example, the ratios of the standard error at $T=50$ to $T=400$ are 7.975, 63.139, and 8.899, respectively. The ratios at $T=100$ to $T=400$ become 3.979, 15.833, and 4.206, respectively. For PS, the ratio of standard errors is comparable to the ratio of the sample size, but for K-averaging, the ratio of standard errors is comparable to the ratio of the sample size squared. These ratios and the ones for other sample size suggest that the estimator using K-averaged converges faster than point sampled data in the order of $\sqrt{T}$, supporting the predictions of Proposition \ref{prop:prop1} that PS is $\sqrt{NT}$ consistent while K-averaging is $\sqrt{N}T$ consistent when $K=T$. Though our theorem does not include LD, the standard errors suggest that its convergence rate is also $\sqrt{NT}$, the same as PS.
\begin{table}[ht!]
\caption{Estimates and Rejection Rates, $(T,N)=(50,50)$}
\label{tbl:table2}
\begin{center}
DGP: $y_{it}= \beta x_{it} + e_{it}$, $e_{it}\sim N(0,3^2)$.
$x_{it}=x_{it-1}+v_{it}+\vartheta v_{it-1}$, $v_{it}\sim N(0,0.5^2)$.
Regression: $\tilde Y_{iK}= c+\beta \tilde X_{iK}+\tilde e_{iK}$
\bigskip
\begin{adjustbox}{width=\textwidth}
\begin{tabular}{lr|rrr|rrrr|rrr}
$\beta$ & $\vartheta$ & \multicolumn{3}{c}{PS($T_1$)} & \multicolumn{4}{c}{KA($T_0,T_1)$} & \multicolumn{3}{c}{LD$(m,T_0+m,T_1-m)$}\\ \hline
& & $T_1=10$ & $T_1=20$ & $T_1=T$ & $(0,T)$ & $(10,20)$ & $(10,40)$ & $(20,40)$ & $m=0$ & $m=2$ & $m=5$\\
\hline
&& \multicolumn{5}{c}{Estimates of $\beta$} \\ \hline
\input table_thm1_est.tex \hline
&& \multicolumn{7}{c}{Rejection Rates of $H_0: \beta=-1$} \\ \hline
\input table_thm1_tstat.tex \hline
\hline
\end{tabular}
\end{adjustbox}
\end{center}
\end{table}
Next, we study the properties of $\hat\beta$ for different configurations of $(\beta,\vartheta)$. As seen in the top panel of Table \ref{tbl:table2}, all time-compressed data precisely estimate $\beta$ for both values of $\vartheta$.
We then vary the true value of $\beta_0$ to evaluate the size of testing $H_0: \beta^0=-1$. The rejection rates are a bit over 5\%, but the normal approximation of the $t$ statistic is reasonably accurate. Power is evaluated at alternatives close to the null value of $\beta^0=-1$. We see from the bottom panel that the power based on K-averaging is much higher than PS and LD. At the alternative of -0.95 (and $\vartheta=0$), for example, K-averaging has a rejection rate of 0.428, compared to a rate of 0.101 for PS and 0.128 for LD, due to their larger standard errors.
\section{More General Specifications}
Proposition \ref{prop:prop1} is built on Assumptions A, B and C. The `meat' of the sandwich variance of the regression in time compressed data is then
\begin{equation}
\label{eq:eqV}
V=\mathbb E_i[ (X_i-\mu_X)^2 \tilde e_{iK}^{2}]= \mathbb E_i[(X_i-\mu_X)^2] \mathbb E_i[\tilde e_{iK}^{2}]+ \text{cov}_i((\tilde X_{iK}-\mu_X)^2,\tilde e^{{2}}_{iK}).
\end{equation}
The covariance term in $V$ does not appear if the model is correctly specified. However,
if this is not the case, $\tilde e_{iK}$ will be replaced by $\tilde e_{iK}^*$, and $\text{cov}_i(\tilde X_{iK},\tilde e^*_{iK})$ may not be zero. The next three subsections
consider cases when $\text{cov}_i(\bar X_{iK},\bar e^*_{iK})\ne 0$ but super-consistency continues to hold. We then consider a case
when $\text{cov}_i(\bar X_{iK},\bar e^*_{iK})=0$ and yet the variance of $\bar e_{iK}^*$ precludes the fastest convergence rate.
\subsection{Relaxing Strict Exogeneity}
We assume that the DGP is the static model $y_{it}=\beta x_{it}+e_{it}$. Assumption B.(i) asserts $\text{cov}_i(\tilde X_{iK},\tilde e_{iK})= \sum_{s}\phi_s \sum_t w(t) \mathbb E_i[u_{it}e_{is}]=0.$
We now allow $\text{cor}(u_{it},e_{is})=
\gamma_{i,t-s}$ with $\sum_h |\gamma_{i,h}|< \infty$. Contemporaneous correlation occurs when $\gamma_{0,i}\ne 0$. When $\gamma_{h,i} \ne 0$ for $h<0$, $u_{it}$ is correlated with future $e_{is}$, while $h>0$ allows for feedback from past values of $e_{it}$ to $u_{it}$.
Now $\tilde X_{iK}=x_{i,T_0}+\sum_s \phi_s u_{is}$ and $\tilde e_{iK}=\sum_t w(t) e_{it}$. Using
$\text{cov}(u_{it},e_{is})=\gamma_{i,t-s} \sigma_i\tau_i$ for any $i$,
\begin{eqnarray*}
\text{cov}(\tilde X_{iK},\tilde e_{iK})&=&\text{cov}\bigg( x_{i,T_0}+ \sum_{s=T_0+1}^{T_1} \phi_s u_{is}, \;
\sum_{t=T_0+1}^{T_1}w(t) e_{it}\bigg)\\
&=&
\sigma_i\tau_i\sum_{s=T_0+1 }^{T_1} \sum_{t=T_0+1}^{T_1} \phi_s w(t) \gamma_{i,s-t}\\
&=& \sigma_i\tau_i \Lambda_K(w,i),
\end{eqnarray*}
where $\Lambda_K(w,i)=\sum_{h=-(K-1)}^{K-1} \gamma_{ih} B_h(w)$ and $ B_h(w)=\sum_{s-t=h} \phi_s w(t)$. The Appendix shows that
$\Lambda_K(w,i) \rightarrow \Gamma_i B_\infty(w), $
where
$\Gamma_i=\lim_{K\rightarrow\infty}\sum_{h=-(K-1)}^{K-1} \gamma_{ih}$ is the long-run cross-covariance, and $B_\infty(w) =\lim_{K\rightarrow\infty} B_h(w)$.
Then \[ \text{cov}(\tilde X_{iK},\tilde e_{iK})\rightarrow \sigma_i\tau_i\Gamma_i B_\infty(w).\]
For $K$ averaging where $w(t)=1/K$ for all $t$, $B_h(w)\rightarrow 1/2$. For point sampling at large $T_1$ where $w(t)=1_{t=T_1}$, $B_h(w)\rightarrow 1$.
With a constant $\tilde b_K$ that depends on the time compression scheme,
\[\mathbb E_i[\text{cov}(\bar X_{iK},\bar e_{iK})]=\mathbb E_i[\sigma_i\tau_i\Gamma_i\tilde b_K]=O(1).\]
Though the average covariance is $O(1)$ due to endogeneity, the cross-section variance is still $O(K)$. In consequence, endogeneity bias vanishes with $K$. Allowing $\text{cov}(u_{it},e_{is})\ne 0$ only inflates variance, in stark contrast to the I(0) case when the estimators would be inconsistent.
Strict exogeneity may also fail because of feedback. Consider the dynamic model
\begin{eqnarray*}
x_{it}&=& x_{it-1}+\kappa y_{it-1}+u_{it}\\
y_{it}&=& \beta_0 x_{it}+\beta_1 x_{it-1}+e_{it}.
\end{eqnarray*}
Then $x_{it}=(1+\kappa \beta_0) x_{it-1}+\kappa \beta_1 x_{it-2}+\kappa e_{it-1}+u_{it}$ is still nearly I(1) if $|\kappa(\beta_0+\beta_1)|$ is small enough to keep $1+\kappa(\beta_0+\beta_1)$ in the $1/T$ neighbourhood of unity. The effective innovation to $x_{it}$ is now $u_{it}^*=\kappa e_{it-1}+u_{it}$. This implies that $\text{cov}(u_{it}^*,e_{i,t-1})=\kappa \tau^2$. A static regression of $\tilde Y_{iK}$ on $\tilde X_{iK}$ now has two biases when $K$ is fixed: one from omitting $\Delta x_{iK}$, and one from violating of strict exogeneity. But signal magnification makes the biases asymptotically negligible. In both cases when strict exogeneity is violated, we have a $\sqrt{N}K$ estimate
of $\beta$ from a static regression using $K$ averaged data, and a $\sqrt{NT_1}$ consistent estimate using data point sampled at $T_1$.
\subsection{Omitting Stationary Dynamics}
In time series regressions when $x$ is I(1), the bias from omitting stationary dynamics may also be of second order. We now show that this is the case in cross-section regressions.
Suppose that $x_{it}=x_{it-1}+u_{it}$ is strictly exogenous, but
\begin{equation}
y_{it}=\alpha y_{it-1} +\beta_0 x_{it}+\beta_1 x_{it-1}+ e_{it}.
\label{eq:eqy1}
\end{equation}
Following \citet{bewley:79}, (\ref{eq:eqy1}) can be rearranged as
\begin{eqnarray*}
(1-\alpha) y_{it}&=& -\alpha \Delta y_{it}+ (\beta_0+\beta_1) x_{it}- \beta_1 \Delta x_{it}+e_{it}\\
y_{it} &=& \bigg(\frac{\beta_0+\beta_1}{1-\alpha}\bigg) x_{it}-\bigg(\frac{\alpha}{1-\alpha} \Delta y_{it} +\frac{\beta_1}{1-\alpha} \Delta x_{it}\bigg)+\frac{e_{it}}{1-\alpha}\\
&=& \beta x_{it}+ e_{it}^*,
\end{eqnarray*}
where $\beta$, often referred to as the long-run multiplier, is
\begin{equation}
\label{eq:LRM} \beta=\frac{\beta_0+\beta_1}{1-\alpha}.
\end{equation}
Since $x$ is I(1), $\beta$ is also the cointegrating coefficient
characterizing the long-run relation between $y$ and $x$. It can be computed as the ratio of two $h$ period impulse response (IRF) to an innovation in $x$ as $h\rightarrow \infty$:
\[ \beta=
\frac{\lim_{h\rightarrow\infty}\mathbb E_t [\partial y_{,it+h}/\partial u_{it}]}
{\lim_{h\rightarrow\infty} \mathbb E_t [\partial x_{i,t+h}/\partial u_{it}]} =
\frac{\lim_{h\rightarrow\infty}\text{IRF}_{u\rightarrow y,h}}{\lim_{h\rightarrow\infty}\text{IRF}_{u\rightarrow x,h}}
\]
The above analysis assumes that $x_{it}$ is non-stationary and strictly exogenous. Suppose that $x_{it}=\rho x_{it-1}+\kappa y_{it-1}+u_{it}$ and let $z_{it}=(x_{it},y_{it})'$.
We have a system of the form $\mathbb B_0 z_{it}=\mathbb B_1 z_{it-1}+ \epsilon_{it}$. Let $\mathbb A=\mathbb B_0^{-1}\mathbb B_1$ and $\mathbb C(L)=(I_2-\mathbb A L)^{-1}\mathbb B_0^{-1}$.\footnote{In this ADL(1,1) model, $\mathbb C(1)=(\mathbb B_0-\mathbb B_1)^{-1}=\frac{1}{D} \begin{pmatrix} 1-\alpha & \gamma \\ \beta_0+\beta_1 & 1-\rho \end{pmatrix}$, $D=(1-\rho)(1-\alpha)-\gamma (\beta_0+\beta_1)$.} Then,
\[ z_t =\mathbb C(L) \begin{pmatrix} u_{it} \\ e_{it} \end{pmatrix}\]
where $\mathbb C=(\mathbb C_0+\mathbb C_1 L+ \ldots \mathbb C_h L^h+\ldots)$, $\mathbb C_h$ is a $2\times 2$ matrix with $\mathbb C_{ij,h}$ in the $(i,j)$-th entry.
The system is non-stationary if the largest eigenvalue of $\mathbb A$ is unity. In this case,
\[ \text{IRF}_{u\rightarrow y,h}=\mathbb C_{21,h}, \quad \text{IRF}_{u\rightarrow x,h}=\mathbb C_{11,h}.\]
If the system is stable, $\mathbb C_{11,\infty}$ and $\mathbb C_{21,\infty}$ are necessarily zero, and we define the long-run multiplier in terms of cumulative impulse responses: $\mathbb C_{21}(1)=\lim_{h\rightarrow\infty} \sum_{k=0}^h= \mathbb C_{21,k}$, and
$\mathbb C_{11}(1)$ is similarly defined. The system implies
\[ \beta=\frac{\beta_0+\beta_1}{1-\alpha} = \begin{cases} \mathbb C_{21}(\infty)/\mathbb C_{11}(\infty)\quad & \text{eig}_{\text{max}}(\mathbb A)=1 \\
\\
\mathbb C_{21}(1)/\mathbb C_{11}(1) \quad & \text{eig}_{\text{max}}(\mathbb A)<1\end{cases}.
\]
Though $\mathbb C_{ij,h}$ depends on all parameters in the system, $\beta$ only depends on the parameters of the equation for $y$ because the other parameters enter $\mathbb C_{21}$ and $\mathbb C_{11}$ through a common determinant that gets canceled out.
As $\beta$ is the long-run multiplier whether or not the system is stable, and whether $x$ is strictly exogenous, estimation of it is of interest. Many methods are available if we take a stand on whether the data are I(1) or I(0). Can we estimate $\beta$ without taking such a stand? A static regression of $y$ on $x$ that omits $\Delta y_{it}, \Delta x_{it-1}$ would have $\mathbb E_i[e^*_{it}x_{it}]\ne 0$. If $x_{it}$ were stationary, a time series or a panel regression would yield inconsistent estimates. But if $x_{it}$ is I(1), a time series regression would yield a super ($T$) consistent estimate because the omitted variable bias is of second order, but the distribution of $\hat\beta$ is non-standard. In terms of K-averaged data,
\begin{eqnarray*}
\bar Y_{iK} &=& \beta \bar X_{iK}-\bigg( \frac{\alpha}{1-\alpha} \Delta y_{iK} +\frac{\beta_1}{1-\alpha} \Delta x_{iK}\bigg) + \frac{\bar e_{iK}}{1-\alpha}\\
&=& \beta \bar X_{iK}- \frac{\eta_{iK}}{1-\alpha}+\frac{\bar e_{iK}}{1-\alpha}
\end{eqnarray*}
where $\Delta y_{iK}=\frac{1}{K}(y_{i,T_1}-y_{i,T_0})$, $\Delta x_{iK}=\frac{1}{K}(x_{i,T_1}-x_{i,T_0})$, and $\eta_{iK}= \alpha \Delta y_{iK}+\beta_1 \Delta x_{iK}$. Importantly, the omitted $\eta_{iK}$ is negligible when $K$ is large.
\begin{proposition}
\label{prop:lrm-I1}
Suppose that $x_{it}$ is I(1), $y_{it}$ is generated by the dynamic model given in (\ref{eq:eqy1}) with $|\alpha|<1$ and Assumption B holds. Let $\hat\beta$ be obtained by estimation of (\ref{eq:S}) using $K$-averaged data.
\begin{itemize}
\item[i] As $N\rightarrow\infty$ with $K$ fixed,
$ \sqrt{N}\bigg(\hat\beta-\beta+\frac{\text{cov}_i(\bar X_{iK},\eta_{iK})}{\text{var}_i(\bar X_{iK})}\bigg)\smash{\mathop{\longrightarrow}\limits^d} N(0,Q_{XX}^{-1}W_K Q_{XX}^{-1})$, where $W_K=\text{var}_i\bigg((\bar X_{iK}-\mu_X) \bar e^*_{iK}\bigg)$ with $\bar e^*_{iK}=\frac{\bar e_{iK}-\eta_{iK}}{1-\alpha}$.
\item[ii] If $x$ is I(1), then for large $K$,
$\sqrt{N}K(\hat\beta-\beta) \smash{\mathop{\longrightarrow}\limits^d} N(0, \mathcal Q_{XX}^{-1} \mathcal W \mathcal Q_{XX}^{-1} )$, where $\mathcal W=\lim_{K\rightarrow\infty } W_K$ and $\mathcal Q_{XX}=\lim_{K\rightarrow\infty} K^{-1} Q_{XX}$.
\end{itemize}
\end{proposition}
In the present setting,
the omitted variable bias for a given $K$ is $\text{bias}=\frac{\text{cov}_i(\bar X_{iK},\bar e^*_{iK})}{\text{var}_i(\bar X_{iK})}$. Since $\bar X_{iK}$ only depends on $\{u_{it}\}$ and $u_{it}\perp e_{is}$ by assumption, bias can only arise through $\text{cov}_i(\bar X_{iK},\eta_{iK})$. As $\bar X_{iK}=O_p(K^{1/2})$ and $\eta_{iK}=O_p(K^{-1})$ in the stationary case, $\text{cov}_i(\bar X_{iK},\eta_{iK})=O(-3/2)$. Upon scaling by $\text{var}_i(\bar X_{iK})=O(K^{-1})$, the bias is $O(K^{-1/2})$. In the I(1) case, $\eta_{iK}=O_p(K^{-1/2})$, but $\bar X_{iK}=O(K^{1/2})$, so $\text{cov}(\eta_{iK},\eta_{iK}=O(1)$
But upon scaling by $\text{var}_i(\bar X_{iK})=O(K)$, the bias is $O_p(K^{-1})$ that vanishes with $K$. We thus have a $\sqrt{N}K$ consistent estimate of the long-run relation $\beta$ as $K$ increases.
Though $\bar e^*_{iK}$ does not contribute to the bias, $\text{var}(\bar e_{iK}^*)>\text{var}(\bar e_{iK})$. Omitting the lags yields a less efficient estimate of $\beta$ as reflected in $W_K$.
It should be noted that $\beta$ is distinct from the long-run effect of a unit shock to $x$, which is
\[ \lim_{h\rightarrow\infty} \mathbb E_t\bigg[\frac{\partial y_{it+h}}{\partial u_{it}}\bigg] = \beta \times \lim_{h\rightarrow\infty}\mathbb E_t\bigg[ \frac{\partial x_{it+h}}{\partial u_{it}}\bigg].\]
If $\beta\ne 0$, the two coincide only if $x_{it}$ is a random walk with innovation variance of one, for in that case, $\mathbb E_t[\frac{\partial x_{i,t+h}}{\partial u_{it}}]=1$ for any $h\ge 0$. In other cases, knowledge about the dynamics of $x_{i}$ is necessary to pin down the long-run effect.
For example, in the IMA(1) case when $\Delta x_{it}= u_{it}+ \vartheta_i u_{it-1}$ and $u_{it}\sim (0,\sigma^2_i)$,
the long-run effect of $u_{it}$ is $\beta(1+\vartheta_i) \sigma_i\ne \beta$.
Consistent estimation of $\beta$ (the long-run multiplier of a change in the level of $x$) is not enough to characterize the long-run effect of an innovation to $x$. We also cannot extrapolate from a relation estimated from variations of medium periodicity to a medium term effect of a shock to $x$.
Any $h$ period effect necessitates estimation of the system $\mathbb B_0 z_{it}=\mathbb B_1 z_{i,t-1}+\epsilon_{it}$, or by local projections. Comparison of the long-run multiplier $\hat\beta$ with the impulse responses such as constructed in \citet{bilal-kanzig:24} should be made with caution.
\subsection{Heterogeneous Coefficients}
The above analysis shows that the strong cross-section variability generated by I(1) data usually accelerates the convergence rate of $\hat\beta$. We now consider a counter example when the strong signal does not accelerate convergence.
Consider heterogeneous coefficient $\beta_i= \beta+ \zeta_i$, where $\mathbb E_i[\zeta_i]=0$, and $\text{var}_i(\zeta_i)=\sigma^2_\zeta$. The model $y_{it}=\beta_i x_{it}+e_{it}$ implies
$ y_{it}=x_{it}\beta+ e_{it}^*$, where $e_{it}^*= \zeta_i x_{it}+e_{it}$, so that
$ \text{cov}(x_{it},e_{is}^*)=\mathbb E[\zeta_i x_{it} x_{is}]\ne 0.$
With $\bar e_{iK}^*=\bar e_{iK}+\zeta_i\bar X_{iK}$,
a static regression in $K$-averaged data yields
\[ \hat\beta = \beta
+ \frac{\text{cov}_i(\zeta_i,\bar X_{iK})}{\text{var}_i(\bar X_{iK})}=\beta+\frac{\mathbb E_i[ \zeta_i(\bar X_{iK}-\mu_X)^2]}{\text{var}_i(\bar X_{iK})}.
\]
If $\zeta_i\perp(\sigma^2_i, x_{i,T_0})$,
the orthogonality condition is satisfied because $\mathbb E_i[\zeta_i]=0$, but the estimator is only $\sqrt{N}$ consistent. This result is best understood in the homoskedastic case when $\sigma^2=\mathbb E_i[\sigma_i^2]$ and $\tau^2=\mathbb E_i[\tau_i^2]$. Then, $\text{var}(\bar e_{iK}^*)= \sigma^2_v \mathbb E_i[\bar X_{iK}^2]+\text{var}(\bar e_{iK})=O(K)$.
$K$-averaging does not achieve noise dilution because parameter heterogeneity scales $\bar e_{iK}^*$ with $\bar X_{iK}$, and any signal magnified by time compression is offset by a magnified noise. Thus, even if $x$ is a random walk with drift, the rate will remain $\sqrt{N}$. \citet{pesaran-smith:95} considered a heterogeneous coefficient model where $x$ is I(1) with a heterogeneous drift. They find that the coefficient estimate from a cross-section regression in full-sample averaged data is consistent. We show that the convergence rate is $\sqrt{N}$, whether or not there is a drift. It should be remarked that point sampling at $T_1$ will also be $\sqrt{N}$ consistent because it also has the problem that $\tilde e_{i,T_1}=e_{i,T_1}+\zeta_i X_{i,T_1}$ scales with $X_{i,T_1}$. The practical implication is that if a researcher wants to allow for heterogeneous coefficients, the usual rate of $\sqrt{N}$ should be expected.
\section{Empirical Issues}
We have so far assumed that $x_{it}$ only has one component that is a stochastic trend. In applications of empirical interest, $x_{it}$ often has additional sources of variation. Subsection 1 considers fixed effects and cross-section dependence. Subsection 2 considers the case when the two components are observed. Subsection 3 considers coexistence of latent permanent and transitory components.
\subsection{Fixed Effects and Cross-Section Dependence}
A common critique of cross-section regressions is the bias from omitted time-invariant variables and cross-section dependence.\footnote{See, for example, \citet{deschenes-greenstone:07}, \citet{mendelsohn-massetti:17}, and \citet{andrews-ecta-05}.} To investigate these issues in our setup, suppose that
\[ y_{it} =\beta x_{it}+ s_i f_t+e_{it},\]
where $x_{it}$ is I(1), $s_i$ is a unit-specific effect, $f_t$ is time variation common across $i$. This interactive fixed effect specification considered in \citet{bai-ecma:09} nests the additive fixed effect as a special case. Few panel data analyses can allow for both I(1) and I(0) common factors. In our setting, I(1) common factors must be controlled for to achieve super-consistent estimates; otherwise, the omitted variations would be the same order as $x$. Then, as in the case of heterogeneous coefficients, the best that can be achieved is $\sqrt{N}$ consistent estimates.
To make progress, let $f_t=\mu_f + f^0_t$ be stationary and $s_i=\mathbb E_i[s_i]+s^0_i$, where $\mu_f=\mathbb E[f_t]$ with $\mathbb E[ f^0_t]=\mathbb E_i[s^0_i]=0$. Because $f^0_t$ is stationary,
\begin{eqnarray*}
s_i\bar f_K
&=& \underbrace{\mathbb E_i[s_i]\mu_f}_{\text{constant}} + s_i^0 \mu_f+\underbrace{s_i \bar f^0_K}_{O(K^{-1/2})}.
\end{eqnarray*}
In this set up, the regression model in $K$-averaged data is
\begin{eqnarray*}
\bar Y_{iK}&=&\beta \bar X_{iK}+ s_i\bar f_K + \bar e_{iK}\\
&=& c + \beta \bar X_{iK} +s^0_i\mu_f + \bar e^*_{iK},
\end{eqnarray*}
where $c$ absorbs the constant term in $s_i\bar f_K$ and $\bar e^*_{iK}=\bar e_{iK}+ s_i\bar f^0_K$.
The omitted variable bias is
\[ \text{bias}=\frac{\text{cov}_i(\bar X_{iK},s_i^0\mu_f)}{\text{var}_i(\bar X_{iK})}=\frac{\mu_f \text{cov}_i(x_{i,T_0},s_i^0)}{\text{var}_i(\bar X_{iK})}.\]
There is trivially no bias when $\mu_f=0$ and $\hat\beta$ continues to be $\sqrt{N}K$ consistent.
More interesting is when $\mu_f\ne 0$, the important special case being $f_t=\mu_f=1$ which makes $s_i$ an individual fixed effect.
If $x_{i0}=0$, strict exogeneity ensures that $\text{cov}_i(x_{i,T_0},s^0_i)=0$ as is required for consistent estimation of $\beta$. The question is at what rate. Now $\bar e_{iK}^*=\bar e_{iK}+s^0_i=O(1)$. While $\bar e_{iK}=O(K^{-1/2})$ and vanishes with $K$, $s^0_i$ cannot be diluted away. So while K-averaging will enjoy some noise dilution for a given $K$, $s^0_i$ will eventually dominate and inflate $V$ in the sandwich variance from $O(1)$ to $O(K)$.\footnote{The situation is different from \citet{andrews-ecta-05}. In Andrews' case, common shocks create cross-section dependence and invalidate the usual standard errors for the $\sqrt{N}$ consistent estimate.} As a consequence, $\hat\beta$ is $\sqrt{NK}$ (instead of $\sqrt{N}K$) consistent. However, the standard errors are valid if $s_i^0$ is independent across $i$. While the convergence rate using $K$-averaged data is downgraded upon losing the benefit of noise dilution, this is not the case if we point sample at $T_1$ because it does not have the noise dilution property anyway. With $\tilde e_{i,T_1}=e_{i,T_1}+s^0_i$ and $\text{var}_i(e_{i,T_1}^*)=O(1)$, the order is the same as $\text{var}(e_{i,T_1})$ when $s^0_i$ was absent, and the convergence rate remains $\sqrt{NT_1}$. The LD differences away the fixed effect to identify $\beta$ from the change rather than level of the non-stationary component, and the convergence rate should be $\sqrt{N}K$, a distinct advantage over $K$-averaging if the fixed effect is truly of the additive type.
The above arguments break down if $s^0_i$ is correlated with $x_{it}$, as strict exogeneity only rules out correlation between $u_{it}$ and $e_{is}$. However, this bias can be mitigated in different ways. First, $s^0_i$ can be controlled by incorporating any variable $z_i$ correlated with $s_i$.
A second possibility is to note that the numerator of the bias $\mu_f\text{cov}(x_{i,T_0},s_i^0)=O(1)$ does not grow with $K$ or $T_0$, implying that the stronger is signal magnification, the faster is convergence. For example, under the $T_{\text{init}}=0$ setup, $\text{var}_i(\bar X_{i,T_1})=O(\mathbb K)$ where $\mathbb K=T_{\text{init}}+K/3$, and $\hat\beta$ is
$\sqrt{N\mathbb K}$ consistent. Similarly, an exogenous drift will also give an estimator with fast enough convergence to reduce the bias from the omitted fixed effect. When the noise dilution mechanism is active, such as in $K$-averaging, omitted fixed effects and cross-section dependence impact the convergence rate of the estimator which depends on the underlying profile of the cross-section variance, but it is at least $\sqrt{N}$ consistent if the model is otherwise well specified.
To illustrate, consider a regression of log consumption on log real GDP, both in per capita terms using data for N=44 countries from the Penn World Table from 1951 to 2023 for $T=73$ periods. In a stochastic growth model, consumption and output share a common stochastic trend associated with productivity, and their ratio is one. But there may be other sources of variation which can only be learned from the data.
As benchmark, we consider three panel estimators for non-stationary data with cross-section dependence captured by common factors $F$ and $\Lambda$ so that $y_{it}=x_{it}\beta+\lambda_i F_t+ e_{it}$. All three estimators have a theoretical convergence rate of $\sqrt{N}\cdot T$. The first is the LSDV, which is pooled least squares estimation in demeaned data. The estimate of 0.927 is consistent if $x$ and $F$ are strictly exogenous but has an asymptotic bias otherwise. The second is the two-step estimator (2sFM) of \citet{bai-kao:06} assuming that $\lambda_i$ and $F_t$ are of dimension two. By first estimating the two factors, a $\hat\beta$ that does not adjust for long-run covariance and endogeneity from possible correlation between $F_t$ and $x_{it}$ is obtained. The third is the continuously updated, bias-corrected (cupBC) estimator of \citet{bkn}. The estimate of 0.950 removes asymptotic bias caused by unit roots, serial correlation, and endogeneity.\footnote{Two other alternatives use cross-section average in observables to capture cross-section dependence.
The CEE-ARDL estimator in \citet{chudik-pesaran-joe:15} requires the factors to be serially uncorrelated and uncorrelated with the regressors. The latter is CEE-DL due to \citet{chudik-etal:16}. With this approach, the errors will be correlated with the regressors when there is feedback, and is sensitive to the stationarity assumption.} The cupBC estimates are serially correlated, but only one of the 44 series has a $\hat\rho_i$ estimated from the autoregressive model $\hat e_{it}=\rho_i \hat e_{i,t-1}+\epsilon_{it}$ that is very close to unity.
\begin{table}[ht!]
\caption{Great Ratio Regression of log Consumption on log GDP, $(T,N)=(73,44)$}
\label{tbl:tbl_app05}
\begin{center}
\begin{tabular}{r|rrrrrrr}
& cupBC & LSDV & 2sFM & PS2 & KA1 & KA2 & LD2 \\ \hline
est & 0.950 & 0.927 & 0.949 & 0.862 & 0.906 & 0.935 & 0.787 \\
se & 0.003 & 0.003 & 0.001 & 0.066 & 0.029 & 0.029 & 0.078 \\
\hline
&& \multicolumn{6}{c}{Simulations} \\ \hline
\input mc_app05_bkn_sample.tex
\hline
\end{tabular}
\end{center}
{\footnotesize Note: cupBC is the continuously updated, bias-corrected estimator of \citet{bkn}. LSDV is least squares estimation in demeaned data. 2sFM is the two-step estimator of \citet{bai-kao:06} that does not correct for endogeneity bias. PS, KA, and LD are cross-section regressions in point-sampled, K-averaging, and long-differencing. KA1 only uses GDP as regressor. PS2, KA2, LD2 additionally use emp/pop to control for cross-section dependence. PS is evaluated at $T_1=2023$ and LD at $(m, m+1,T-m)$ with $m=2$.}
\end{table}
The first factor estimated from cupBC is cyclical and appears stationary, while the second appears to be flat until around 1990 when a trend emerges. The trend may be due to the fact that our sample includes developing countries which have seen increased participation rate. Let $x_{1,it}$ be log GDP, $x_{2,it}= \text{emp}_{it}/\text{pop}_{it}$, where emp and pop are employment and population data. Let KA2 be estimates using K-averaged data in a cross-section regression with $\bar X_{1,iK}$ and $\bar X_{2,iK}$ as regressors.
The full sample KA2 estimate of 0.935 is close to cupBC of 0.950, while the point sampled estimate (PS2) at $T$ is 0.862, and the LD2 estimate with $(m,T_a,T_b)=(2,m+1,T-m)$ is 0.787.
The standard error of KA2 is 9.66 times larger than that of cupBC, and $\sqrt{T}$ in this example is 8.54. This suggests that there are omitted fixed effects that slowed convergence of the estimates using K-averaged data from the fastest possible rate of $\sqrt{N}\cdot T$ to $\sqrt{NT}$, as noted above.
\begin{figure}[!ht]
\caption{Regression of Log Consumption on Log GDP}
\label{fig:app05}
\begin{center}
\includegraphics[width=6.0in,height=2.0in]{do_app05_seg.png}
\end{center}
{\footnotesize Note: The blue (K-averaged) and green (LD) lines are estimates for six choices of ($T_0,T_1$) pairs. The red line is the estimate from point sampling at every $t$, and its one-standard error confidence interval in dotted red.}
\end{figure}
If the regression model is well-specified, $\hat\beta$ should be fairly stable with respect to $K$, and in our experience, the sensitivity of the K-averaging estimates to $K$ is a useful diagnostic of the adequacy of the model. Without $\bar X_{2i,K}$, the KA1 estimates are quite sensitive to the choice of $T_0$ and $T_1$.
Figure \ref{fig:app05} plots the estimates $(T_0,T_1)=(\kappa,T-\kappa)$ pairs for $\kappa$=2:5:30 and their standard errors. We see that the KA2 estimates (in blue) are stable with respect to $K=T_1-T_0$, but the point-sampled estimates PS2 (in red) display significant variability after 2000. Though within two standard errors of panel estimates, PS2 at $T_1=2023$ is only 0.862. The LD2 estimates are also sensitive to the choice of $T_0$ and $T_1$. With $m=2$, the LD2 estimate from the longest sample is 0.787, though much lower, it is roughly within two standard errors of KA2 and the panel estimates.
To be confident of the estimates, a calibrated monte-carlo exercise with 10,000 replications is conducted by generating data using the cupBC estimate of 0.950 with the two estimated factors fixed. The corresponding loadings, the individual fixed effect, and the autocorrelation coefficients $\hat\rho$ of the cupBC residuals are resampled. For each $i$, synthetic AR1 errors are generated using $\hat\rho_i$ and noise $\epsilon_{it} \sim N(0,\sigma^2_{\hat\epsilon,i}$). The results are shown in the bottom panel of Table \ref{tbl:tbl_app05}. The exercise shows that the panel estimators are more efficient, as should be the case, because of explicit modeling of the common factors and biases from different sources. The cross-section estimates perform well considering that no correction is made, and standard normal inference is possible. The LD2 and PS2 estimates are closer to KA2 and the panel estimates than in the data. Further investigation reveals that if we only resample $\Lambda_1$ but not $\Lambda_2$ (corresponding to the non-stationary factor), the mean PS2 and LD2 estimates in the monte carlo are close to the point estimates from the data. This points to the sensitivity of both estimators to the choice of $T_0$ and $T_1$ for the data being analyzed, and more generally that differencing may not adequately control for common variations of the interactive type.
\subsection{Fanning-in vs Fanning-out}
This subsection considers the case when $x$ is the sum of two observed variables with distinctive properties.
Consider the dependency ratio which is the sum of the population under 16 ('young') and over 65 ('old') relative to the working age population. Using data from the 110 countries in the World Bank database from 1960-2022, the top panel of Figure \ref{fig:young} shows that `old' is non-stationary; the time series data have been steadily increasing, while its cross-section variance fans out. The bottom panel shows that while the time series data of the `young' are also non-stationary, the cross-section variance fans out until the turn of this century. The dependency ratio and population growth exhibit a similar feature as `young'. A possible explanation is the global decline in fertility which creates convergence across units. The cross-section variance of the dependency ratio thus has a fanning-out component (due to the 'old') and a fanning-in (due to the 'young') component.
\begin{figure}[!ht]
\caption{Share of Old and Young in Working Population}
\label{fig:young}
\includegraphics[width=7.0in,height=1.500in]{do_app06pwtwb_old.png}
\medskip
\includegraphics[width=7.0in,height=1.50in]{do_app06pwtwb_young.png}
\end{figure}
A high dependency ratio has been suggested as a possible explanation for higher inflation and a drag on growth.\footnote{See, for example, \citet{juselius-takats:21} for inflation. \citet{gordon:16} argues that an aging population creates a structural imbalance between high savings and low investment. A different view is given in \citet{acemoglu-restrepo:17}.} The mixed evidence could be due to differences in the choice of the sample, as the relation between economic and demographic variables could be different for countries in different stages of economic development. We use a sub-sample of 35 countries with average real per-capita GDP in the past 10 years over 10,000 USD, and with maximum inflation in the sample under 100 percent. In this data, the cross-section variance of the `young' fans-in over the entire sample, distinctly different from the fanning out cross-section variance of `old'. If we use the dependency ratio as regressor, we find that the estimate is positive in the inflation regression and significant but varies significantly as $K$ increases. The coefficient in the growth regression is more stable but is not significant. Because the dependency ratio has two components with different properties, we explore their distinctive economic impact.
\begin{table}
\caption{Growth/Inflation Regressions on Demographic Variables: K-averaging}
\label{tbl:demo}
\begin{center}
\begin{tabular}{ll|llll|llll}
$T_0$ & $T_1$ & \multicolumn{4}{c}{inflation} & \multicolumn{4}{c}{growth} \\ \hline
&& $\hat\beta_{old}$ & $t_{old}$ & $\hat\beta_{young}$ & $t_{young}$
& $\hat\beta_{old}$ & $t_{old}$ & $\hat\beta_{young}$ & $t_{young}$ \\ \hline
\hline
1963 & 2020 & 0.289 & 1.495 & 0.406 & 1.508 &-0.154 & -2.338 & -0.048 & -1.542 \\
1968 & 2015 & 0.302 & 1.548 & 0.465 & 1.558 &-0.172 & -2.794 & -0.048 & -1.653 \\
1973 & 2010 & 0.333 & 1.619 & 0.545 & 1.574 &-0.176 & -2.885 & -0.055 & -1.928 \\
1978 & 2005 & 0.354 & 1.740 & 0.623 & 1.597 &-0.164 & -2.512 & -0.062 & -1.939 \\
1983 & 2000 & 0.281 & 1.693 & 0.603 & 1.480 &-0.152 & -1.849 & -0.065 & -1.665 \\
\hline
\end{tabular}
\end{center}
{\footnotesize Regressions of Inflation (left) and Growth (right) on `old' and 'young' for a subsample of 35 countries, 1960-2022. The 'old' and `young' are from the World Bank database, and the economic variables are from the Penn World Table.}
\end{table}
Table \ref{tbl:demo} reports results from regressions of inflation (left) and growth (right) on the demographic variables for $(T_0, T_1)=(\kappa,T-\kappa)$ pairs with $\kappa$=2:5:25.
For inflation, the full sample estimate of the coefficient on `old' is 0.289, and the coefficient on `young' is 0.406, but neither is statistically significant at the 10\% level. For the growth regression, the full sample estimate of the coefficient on 'old is -0.154 with a $t$ statistic of -2.338 and all subsample estimates are similar. The coefficient on `young' is -0.048 and the subsample estimates are similar, but the estimates are only marginally significant. The analysis suggests that the effect of the dependent population on inflation is positive but not statistically well determined, while there is a negative effect of both `old' and `young' on growth.\footnote{For the inflation regression, the LD estimate for the longest sample is -0.194 with a $t$ statistic of -2.219, and estimates of the shorter subsamples are not statistically significant. The LD longest sample estimate of `old' in the growth regression -0.184 with a $t$ statistic of -4.421, but the subsample estimates are more variable, ranging from -0.051 to -0.451.} A possible explanation for why the estimate on `old' is better determined is that the data for 'old' has more variability because its cross-section variance fans out. When there are two underlying components in the covariate, using two covariates instead of the sum can be informative.
\subsection{Permanent and Transitory Components}
The dependency ratio is a sum of two components that we observe.
Suppose now $x_{it}=\mu_{it}+v_{it}$ is a sum of two latent components, where $\mu_{it}=\mu_{it-1}+u_{it}$ is non-stationary and $v_{it}$ is a mean-reverting process. As shown earlier, $x_{it}$ can be represented as an IMA(1) process with moving-average parameter $\vartheta$.
If $\mu_{it}$ and $v_{it}$ have the same effect on $y_{it}$, the presence of $v_{it}$ would not change the analysis because $\Delta x_{it}=u_{it}$ already allows for the possibility that $u_{it}$ can be serially correlated. Nonetheless, as seen in Table \ref{tbl:csvar}, the cross-section variance will be small if the variability from $\mu_{it}$ is small.
When the fanning-out effect is limited, regressions in time-compressed I(1) data will not benefit from signal magnification. In that case, omitted serial correlation and fixed effect will no longer be of second-order importance when the data only permit $\sqrt{N}$ consistent estimation.
A second issue arises if $\mu_{it}$ and $v_{it}$ have distinctive impact of $y$. Consider
\[y_{it}=\beta_\mu \mu_{it}+\beta_v v_{it}+e_{it}.\]
As $\text{cov}(x,y)=\beta_\mu \text{var}(\mu)+ \beta_v\text{var}(v)$,
the estimand ($\beta_x$) in a regression of $y$ on $x$ will be a variance weighted average of $\beta_\mu$ and $\beta_v$. It is as if the regression error $e_{it}^*= (\beta_v-\beta_\mu) v_{it}+e_{it}$ has a measurement noise. The problem is analogous to the inability to identify the effect of permanent income on consumption from income data. See, for example, \citet{cochrane-94}. With point-sampling at $T_1$, the weight on $\beta_\mu$ would depend exclusively on the cross-section volatility of $v_{i,t}$ at $t=T_1$. Here, a regression in
$K$-averaged data helps because the estimand is
\begin{eqnarray*}
\beta_x=\frac{\text{cov}_i(\bar Y_{iK},\bar X_{iK})}{\text{var}_i(\bar X_{iK})}=\frac{\beta_\mu \text{var}_i(\bar \mu_{iK})+\beta_v \text{var}_i(\bar v_{iK})}{\text{var}_i(\bar \mu_{iK})+\text{var}_i(\bar v_{iK})}.
\end{eqnarray*}
In terms of regression noise, $\bar e_{iK}^*=(\beta_v-\beta_\mu)\bar v_{iK}+\bar e_{iK}=O(K^{-1})$, and in terms of signal, $\text{var}_i(\bar \mu_{iK})=O(K)$. If $K$ is small, OLS will estimate a weighted average of $\beta_\mu$ and $\beta_v$. But $\beta_x\rightarrow \beta_\mu$ as $K\rightarrow \infty$ if $\text{var}_i(\bar \mu_{iK})=O(K)$. A larger $K$ better isolates $\beta_\mu$ by increasing the permanent-to-transitory ratio $\sigma^2_{\bar \mu_K}/\sigma^2_{\bar v_K}$.
\begin{table}[ht!]
\caption{Simulations Using the Local-Level Model}
\label{tbl:table3}
\begin{center}
DGP: $y_{it}= \beta_{\mu} \mu_{it} + \beta_v v_{it} + e_{it}$, $(\beta_\mu,\beta_v)=(-1,-2.5)$,
Cross-section regression: $\tilde Y_{iK} = \beta_x \tilde X_{iK} + \tilde e_{iK}$,
$(T,N)=(50,50)$, $(T_0,T_1,T_2)=(10,20,40)$
\begin{adjustbox}{width=\textwidth}
\begin{tabular}{lll|rr|rrr|rrrr|rrr}
$\sigma_u$ & $\rho $ & $\vartheta$ & \multicolumn{2}{c}{Oracle} & \multicolumn{3}{c}{PS} & \multicolumn{4}{c}{KA} & \multicolumn{3}{c}{$LD_K(m,T_0,T_2)$}\\ \hline
& & & $\hat\beta_\mu^*$ & $ \hat\beta_w^*$ &$T_1$ & $T_2$ & $T$ & $(1,T)$ & $(T_0,T_1)$ & $(T_0,T_2)$ & $(T_1,T_2)$ & $m=0$ & $m=2$ & $m=5$\\ \hline
\input table3_locallevel_est.tex
\input table3_locallevel_se.tex
\end{tabular}
\end{adjustbox}
\end{center}
\end{table}
To illustrate this point, we conduct a monte-carlo exercise with 10,000 draws of data from the local-level model for different values of $\sigma_u$ and $\rho$ with $(\beta_\mu,\beta_v)=(-1.0,-2.5)$. The estimates of $\beta_x$ and corresponding standard errors are reported in Table \ref{tbl:table3}. A panel regression using $\mu_{it}$ and $w_{it}$ as covariates is infeasible because these variables are not observed by the econometrician, but they are observed in simulations. We label these estimates as \textsc{oracle}. Table \ref{tbl:table3} shows that $\hat\beta_x$ is often close to $\beta_\mu$ when $\vartheta$ is away from -1, but is contaminated by $\beta_w$ when $\vartheta$ is close to -1. K-averaging is more capable of producing estimates $\hat\beta_x$ that are closer to $\beta_\mu$ than LD or PS when $\vartheta\ne -1$ .
Both problems can be relevant to regressors such as temperature if we associate $\mu_{it}$ with the slowly evolving warming trend and $v_{it}$ with regular temperature fluctuations, and itt seems plausible that a warming temperature trend $\mu_{it}$ may affect economic outcome $y_{it}$ differently from regular temperature fluctuations $v_{it}$.
Figure \ref{fig:ustemp} plots the temperature data (in Celsius) for the 48 states in the U.S.\footnote{The latitude data is based on USGS Geographic Names Information System (GNIS), utilizing the North American Datum of 1983 (NAD 83) aligned with standard WGS 84 latitude-longitude configurations.
} There is a slight upward trend in the time series, and yet the cross-section variance hardly fans out. The limited fanning out now makes it even more challenging to identify $\beta_\mu$. Unless $\beta_\mu=\beta_v$, a regression of $y$ on $x$ may not return a meaningful parameter estimate, and even time compressed data may not help.
\begin{figure}[ht!]
\caption{Temperature in Celsius for 48 States in the U.S.}
\label{fig:ustemp}
\hspace*{-1.0in}
\includegraphics[width=8.0in,height=2.50in]{do_app01_temp.png}
\end{figure}
To assess the empirical relevance of the problems, we use output growth as economic outcome; i.e., $y_{it}=100 \Delta \log \text{GSP}_{it}$, where GSP is real state product per capita. As noted in the consumption example, it is useful to check if the estimates using K-averaged data are stable with respect to $K$. We first estimate the linear model $\bar Y_{iK}=c+ \beta_1 \bar X_{iK}+ $ error and find that the estimates decline as $K$ increases. Adding latitude of the state capital stabilizes the estimates, reinforcing what has been suggested in the literature that cross-section regressions can be sensitive to omitted variable bias.
Next, we consider possible non-linear effects by adding a quadratic term:
\[\bar Y_{iK}= b_0 + \beta_1 \bar X_{iK} + \beta_2 \bar X^2_{iK} + \text{latitude}_i+\text{ error}.
\]
If $x_{it}=x_{i0}+\sum_{s=1}^t u_{is}$ is a random walk, then $\mathbb E[x_{it}^2]=\mathbb E[x_{i0}^2]+t \sigma^2_i$ but $x_{it}^2$ is not I(2) because it remains non-stationary after differencing.\footnote{In particular, $\text{var}(\Delta x_{it}^2)=4x_{i,t-1}^2\sigma^2_i+2\sigma_i^4=O(t)$. \citet{ng:08} refers to $x_{it}^2$ as a heteroskedastic random walk; see also \citet{rico-gonzalo:14}.} Inference would be difficult for time series regressions. Interestingly, in this data, $ \text{var}_i(x_{it}^2)$ fans out even though $\text{var}_i(x_{it})$ does not. With time compressed data, standard normal inference is possible without knowing the exact order of integration.
Note, however, that $\bar X_{iK}^2$ is the average of $x_{it}^2$, not $(\bar X_{ia})^2$ as in \citet{burke-emerick:16}. It should also be noted that once the non-linear term is present, the estimates using K-averaged data are similar with or without 'latitude and stable with respect to $K$.
\begin{figure}[ht!]
\caption{Regressions of Growth on Temperature: 48 States}
\label{fig:climate}
\medskip
\includegraphics[width=6.250in,height=3.50in]{do_app01_seg.png}
{\footnotesize See notes to Figure 2.}
\end{figure}
Figure \ref{fig:climate} plots estimates from the non-linear model for different $(T_0,T_1)=(\kappa,T_1-\kappa)$ pairs with $\kappa$=2:5:30. The K-averaging estimates (in blue) of $\beta_1$ are negative while the $\beta_2$ estimates are positive. However, the $t$ statistics for full-sample averaging are -1.301 and 1.236, respectively, suggesting that both the linear and non-linear terms are not statistically different from zero. Evidence for a significant long-run relation between growth and temperature from the sub-sample estimates is even weaker. In this application, the LD estimates (in green) range from positive to negative, but they are also not well determined. Thus, even with methodologies that target the non-stationary variations, $\beta_\mu$ is not well determined in the data.
The point sampled estimates help to shed light on the results in this setting. As seen in Figure \ref{fig:climate}, there are significant fluctuations in the PS estimates (in red) over time, suggesting that the stationary variations dominate the data, consistent with the lack of fanning out in the cross-section variance of temperature. When the non-stationary variation is so weak, the cross-section estimate of $\beta_x$ will be close to $\beta_v$ which a panel regression with time and individual fixed effect targets. Similarity of the cross-section (long-run) and panel data (short-run) estimates have been interpreted as lack of adaptation in the literature. But it could also be that the data does not have enough trending variations to precisely estimate its long-run relation between temperature and growth.
\section{Conclusion and Discussion}
This paper is built on the result that the cross-section variance of a class of time-compressed I(1) data will fan out, and the magnified signal can be exploited to yield (super-)consistent and asymptotically normal estimates. The fastest convergence rate obtains when the compression scheme not only amplifies signal, but also dilutes the stationary regression noise to render endogeneity and misspecification bias of second order. We investigate the long-run relation between consumption and income, between inflation/growth and demographic variables, and between temperature and growth. While the first two examples find estimates that are stable with respect to $K$, the last example finds that temperature data are dominated by short-term fluctuations, making it difficult even for time compression schemes to isolate the effect of the trending variations.
\clearpage
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