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Testing Hypotheses About Ratios of Linear Trend Slopes in Systems of Equations with a Focus on Tests of Equal Trend Ratios
Keywords: Trend Stationary, Instrumental Variables Estimation, HAC Estimator, Fixed-b Asymptotics, Amplification Ratio, Long Run Variance
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This paper analyzes estimation and inference methods for parameters that represent ratios of pairs of linear trend slopes in a system of linear trending time series with covariance stationary fluctuations around the trend. The proposed methods extend the results of vogelsang-nawaz-JTSA where the focus was on a single trend ratio parameter, and it was shown that instrumental variables (IV) estimation using time as an instrument was preferred to ordinary least squares (OLS) of the relevant estimating equation. Here, there is a vector of trend ratio parameters estimated by IV. Inference focuses on hypotheses that represent linear restrictions across trend ratio parameters. Much of the intuition in vogelsang-nawaz-JTSA for the single pair case extends to the multi-pair case especially with respect to the importance of the magnitude of trend slopes relative to noise (variation/fluctuations) around the trends for estimation and inference. Throughout the paper, magnitudes of trends slopes are always interpreted relative to the magnitude of the noise.
Detailed attention is given to the special case where an empirical practitioner wants to test the equality of trend ratio parameters between two pairs of series. This special case is directly relevant for the study of amplification ratios in the empirical climate literature; \nocite{santer2005amplification}Santer et al (2005), \nocite{thorne2007tropical}Thorne et al (2007), \nocite{klotzbach2009alternative}\nocite{klotzbach2010correction}Klotzbach et al (2009, 2010), \nocite{christy2010observational}Christy et al (2010), po2012discrepancies, vogelsang-nawaz-JTSA , VMCS_2026 (hereafter VMCS26), and references in those papers.
The amplification ratio is the ratio of a temperature trend in the troposphere of the earth relative to a temperature trend at the surface of the earth. A key assessment of theoretical climate models is determining whether estimated amplification ratios of observed temperature series are aligned with estimated amplification ratios of model generated temperatures. The null hypothesis of interest is equality of trend ratios between two pairs of temperature series: one pair for observed temperatures and a second pair for model generated temperatures. Whereas the previous literature treats the amplification ratio computed for model generated temperatures as fixed (ignoring that it is estimated), the methods developed in this paper treat both the observed and model amplification ratios as estimators. Inference takes into account the joint sampling distributions of both estimated amplification ratios. VMCS26 use these methods to assess the alignment of amplification ratios in CMIP6 model generated temperature series with amplification ratios in observed temperature series. VMCS26 find systematic misalignments especially for amplification ratios in the lower troposphere in which case the models exhibit substantially higher amplification (more warming relative to the surface) than in observed temperatures. In the present paper amplification ratios are compared across five sets of observed temperatures.
The remainder of the paper is organized as follows: Section 2 lays out the system of estimating equations used to estimate trend ratios of pairs of linear trending time series. The asymptotic properties of the IV estimator of the trend slope ratios is provided. As shown by vogelsang-nawaz-JTSA, IV estimation is used rather than OLS because of systematic correlation between the regressors and regression errors. Section 3 provides a framework for testing linear restrictions of trend ratios across equations in the system. Test statistics are configured to be robust to serial correlation in the fluctuations of the time series around their trends as well as correlation across time series. Tests of equal trend ratios between pairs is obtained as a special case. When the trend slopes are small relative to the variation of the time series around their trend, the IV estimators, and tests built on them, can behave very differently than when trend slopes are large. To help offset this sensitivity to the magnitude of the trend slopes, Section 4 explores an alternative approach to inference for tests of equal trends that is labeled the "product approach". The idea is related to the linear-in-trend slopes inference method (\nocite{fieller1954}Fieller 1954) used by vogelsang-nawaz-JTSA for a single trend ratio. Unlike the linear in trend slopes approach, the product approach is not fully robust to very small, or even zero, trend slopes. However, as the finite sample simulations show in Section 5, the product approach for testing equal trend slopes is often less sensitive to very small trend slopes relative to tests based on the IV estimators. The simulations suggest that while the product approach can be more robust to very small trend slopes, IV based tests tend to be less sensitive to strong autocorrelation (tendency to over-reject under the null hypothesis is less). Section 6 provides some practical recommendations for empirical researchers. Section 7 uses the proposed tests of equal trend ratios to compare amplification ratios across five sets of observed temperatures for the tropics of the earth. Section 8 concludes, and proofs are provided in an appendix.
The setup consists of $i=1,2,\ldots,n$ pairs of univariate linear trending time series, $y_{1t}^{(i)}$ and $y_{2t}^{(i)}$, given by
where $u_{1t}^{(i)}$ and $u_{2t}^{(i)}$ are mean zero covariance stationary processes and $t=1,2,\ldots,T$. The parameters of interest are the $n$ ratios of trend slopes between each pairs of series given by \[ \theta^{(i)}=\frac{\beta_{1}^{(i)}}{\beta_{2}^{(i)}}, \] where $\beta_{2}^{(i)}\neq0$. Using simple algebra from vogelsang-nawaz-JTSA, estimating equations for the ratios can be derived as
where \[ \delta^{(i)}=\mu_{1}^{(i)}-\theta^{(i)}\mu_{2}^{(i)},\text{ \ \ \ \ } \epsilon_{\theta t}^{(i)}=u_{1t}^{(i)}-\theta^{(i)}u_{2t}^{(i)}. \] Throughout the paper, the time series are assumed to be covariance stationary around their respective linear trends and that sufficient weak dependence holds so that a functional central limit theorem (FCLT) holds for the $2n\times1$ vector: \[ \mathbf{U}_{t}=\left[
\right] , \] where $\mathbf{U}_{1t}=\left[ u_{1t}^{(1)},u_{1t}^{(2)},\ldots,u_{1t} ^{(n)}\right] ^{\prime}$ and $\mathbf{U}_{2t}=\left[ u_{2t}^{(1)} ,u_{2t}^{(2)},\ldots,u_{2t}^{(n)}\right] ^{\prime}$. Specifically, it is assumed that
where $r\in\lbrack0,1]$ and $[rT]$ is the integer part of $rT$. The elements of the $n\times1$ vectors of Brownian motions, $\mathbf{B}_{\mathbf{u}1}(r)$ and $\mathbf{B}_{\mathbf{u}2}(r)$, are given by \[ \mathbf{B}_{\mathbf{u}1}(r)=\left[ B_{u1}^{(1)}(r),B_{u1}^{(2)} (r)\ldots,B_{u1}^{(n)}(r)\right] ^{\prime},\text{ \ \ \ \ }\mathbf{B} _{\mathbf{u}2}(r)=\left[ B_{u2}^{(1)}(r),B_{u2}^{(2)}(r)\ldots,B_{u2} ^{(n)}(r)\right] ^{\prime}. \] The vector of Brownian motions, $\mathbf{B}_{\mathbf{u}}\mathbf{(r)}$, can be written as $\mathbf{\Lambda}_{\mathbf{u}}\mathbf{W}_{\mathbf{u}}(r)$ where $\mathbf{W}_{\mathbf{u}}(r)$ is a $2n\times1$ vector of independent standard Wiener processes and $\mathbf{\Omega}_{\mathbf{u}}=\mathbf{\Lambda }_{\mathbf{u}}\mathbf{\Lambda}_{\mathbf{u}}^{\prime}$ is the long run variance of $\mathbf{U}_{t}$. It is\ not assumed that $\mathbf{\Omega }_{\mathbf{u}}$ is diagonal allowing for correlation across elements of\ $\mathbf{U}_{t}$ (within and across pairs). In addition to ((ref)), it is assumed that $\mathbf{U}_{t}$ is ergodic for the first and second moments.
Stacking the individual estimation equation errors, $\epsilon_{\theta t} ^{(i)}$, gives the $n\times1$ vector, $\scalebox{1.5}{$\bm{\epsilon}$}_{\theta t}$, that can be written as \[ \scalebox{1.5}{$\bm{\epsilon}$}_{\theta t}=\mathbf{U}_{1t}-\mathbf{D} _{\mathbf{\theta}}\mathbf{U}_{2t}, \] where \[ \mathbf{\theta=}\left[ \theta^{(1)},\theta^{(2)},\ldots,\theta^{(n)}\right] ^{\prime}, \] and $\mathbf{D}_{\mathbf{\theta}}$ is an $n\times n$ diagonal matrix with $i^{th}$ diagonal elements $\theta^{(i)}$. Notice that $\scalebox{1.5}{$\bm{\epsilon}$}_{\theta t}$ can be written as a linear function of $\mathbf{U}_{t}$ through the relationship \[ \scalebox{1.5}{$\bm{\epsilon}$}_{\theta t}=\left[ \mathbf{I}_{n} ,-\mathbf{D}_{\mathbf{\theta}}\right] \mathbf{U}_{t}, \] where $\mathbf{I}_{n}$ is an $n\times n$ identity matrix. It immediately follows from ((ref)) that
where $\mathbf{W}_{\mathbf{\epsilon}}(r)$ is an $n\times1$ vector of independent Wiener processes and \[ \mathbf{\Omega}_{\mathbf{\epsilon}}=\mathbf{\Lambda}_{\mathbf{\epsilon} }\mathbf{\Lambda}_{\mathbf{\epsilon}}^{\prime}=\left[ \mathbf{I} _{n},-\mathbf{D}_{\mathbf{\theta}}\right] \mathbf{\Lambda}_{\mathbf{u} }\mathbf{\Lambda}_{\mathbf{u}}^{\prime}\left[ \mathbf{I}_{n},-\mathbf{D} _{\mathbf{\theta}}\right] ^{\prime}, \] is the long run variance of $\scalebox{1.5}{$\bm{\epsilon}$}_{\theta t}$.
The trend slope ratios $\theta^{(i)}$ can be estimated using the estimating equations ((ref)). While ordinary least squares (OLS) applied equation by equation for each $i$ would seem natural, vogelsang-nawaz-JTSA showed that OLS applied to ((ref)), for a given $i$, yields a biased estimator of $\theta^{(i)}$. The bias is caused by correlation between $y_{2t}^{(i)}$ and $\epsilon_{\theta t}^{(i)}$ through the common term $u_{2t}^{(i)}$. Instead, vogelsang-nawaz-JTSA recommend using instrumental variables (IV) estimation using $t$ as the instrument for $y_{2t}^{(i)}$. The IV estimators are defined as
where $\overline{y}_{1}^{(i)}=T^{-1}\sum_{t=1}^{T}y_{1t}^{(i)}$, $\overline {y}_{2}^{(i)}=T^{-1}\sum_{t=1}^{T}y_{2t}^{(i)}$ and $\overline{t}=T^{-1} \sum_{t=1}^{T}t$ are sample averages. Standard algebra gives the relationship \[ \widehat{\theta}^{(i)}-\theta^{(i)}=\left( {\displaystyle\sum\limits_{t=1}^{T}} (t-\overline{t})(y_{2t}^{(i)}-\overline{y}_{2}^{(i)})\right) ^{-1} {\displaystyle\sum\limits_{t=1}^{T}} (t-\overline{t})\epsilon_{\theta t}^{(i)}. \] Notice that the IV estimator can be equivalently written as \[ \widehat{\theta}^{(i)}=\frac{\widehat{\beta}_{1}^{(i)}}{\widehat{\beta} _{2}^{(i)}}, \] where $\widehat{\beta}_{1}^{(i)}$ and $\widehat{\beta}_{2}^{(i)}$ are the OLS estimators of $\beta_{1}^{(i)}$ and $\beta_{2}^{(i)}$ based on regressions ((ref)) and ((ref)):
Stack the $\widehat{\theta}^{(i)}$ into the vector \[ \widehat{\mathbf{\theta}}\mathbf{=}\left[ \widehat{\theta}^{(1)} ,\widehat{\theta}^{(2)},\ldots,\widehat{\theta}^{(n)}\right] ^{\prime}. \] Using ((ref)), $\widehat{\mathbf{\theta}}$ can be written as \[ \widehat{\mathbf{\theta}}\text{ }\mathbf{=}\text{ }\widehat{\mathbf{D}} _{2}^{-1}\sum_{t=1}^{T}\left( \mathbf{y}_{1t}-\overline{\mathbf{y}} _{1}\right) \left( t-\overline{t}\right) , \] where $\mathbf{y}_{1t}=\left[ y_{1t}^{(1)},y_{1t}^{(2)},\ldots,y_{1t} ^{(n)}\right] ^{\prime}$ and $\widehat{\mathbf{D}}_{2}$ is an $n\times n$ diagonal matrix with $i^{th}$ diagonal element given by $\sum_{t=1}^{T}\left( t-\overline{t}\right) \left( y_{2t}^{(i)}-\overline{y}_{2}^{(i)}\right) $. Standard calculations give
which holds as long as the trend slopes are nonzero. When trends slopes are zero, $\mathbf{\theta}$ is not defined and it follows that
For the rest of the paper, the focus is on the IV estimator, $\widehat{\mathbf{\theta}}$, and tests of linear restrictions regarding $\mathbf{\theta}$.
The following Theorem gives the asymptotic properties of $\widehat{\mathbf{\theta}}$ under the assumption that the FCLT ((ref)) holds. The asymptotic limit of $\widehat{\mathbf{\theta}}$ depends on the magnitude of the trend slope parameters, $\beta_{1}^{(i)},\beta_{2}^{(i)}$ relative to the variation in the random components (noise), $u_{1t}^{(i)}$ and $u_{2t}^{(i)}$.
For Cases 1 and 2 the limits given in Theorem 1 are multivariate versions of the limits obtained by vogelsang-nawaz-JTSA and are identical to the limits in vogelsang-nawaz-JTSA when $n=1$. In Case 1, $\widehat{\mathbf{\theta}}$ consistently estimates $\mathbf{\theta}$ and the precision of $\widehat{\mathbf{\theta}}$ depends on the magnitudes of the trends slopes in the denominator series. In Case 2, $\widehat{\mathbf{\theta} }$ becomes inconsistent. This is not surprising because the case of very small slopes means that the trend component of $y_{2t}^{(i)}$ is dominated by the noise, $u_{2t}^{(i)}$, in which case $t$ is a weak instrument (Staiger and Stock 1997\nocite{staiger-stock}) for $y_{2t}^{(i)}$. When all trend slopes are zero (Case 3), trend ratios are not defined and $\widehat{\mathbf{\theta} }$ converges to a random vector that depends on $\mathbf{B}_{\mathbf{u} }\mathbf{(r)}$. It is obvious that for very small and zero slopes, inference will be affected by the different behavior of $\widehat{\mathbf{\theta}}$.
Suppose one is interested in testing linear restrictions across the trend slopes, $\theta^{(i)}$, using the IV estimators $\widehat{\theta}^{(i)} $. More formally, consider testing the null hypothesis \[ H_{0}:\mathbf{R\theta=r,} \] against the alternative \[ H_{1}:\mathbf{R\theta\neq r,} \] where $\mathbf{R}$ is a known $q\times n$ matrix with $rank(\mathbf{R)=}q$ and $\mathbf{r}$ is a known $q\times1$ vector.
Using the Case 1 limit from Theorem 1, the Wald statistic for testing $H_{0}$ against $H_{1}$ is given by \[ Wald_{IV}=\left( \mathbf{R}\widehat{\mathbf{\theta}}-\mathbf{r}\right) ^{\prime}\left[ \mathbf{R}\widehat{\mathbf{V}}_{IV}\mathbf{R}^{\prime }\right] ^{-1}\left( \mathbf{R}\widehat{\mathbf{\theta}}-\mathbf{r}\right) \] where \[ \widehat{\mathbf{V}}_{IV}=\left( \sum\nolimits_{t=1}^{T}(t-\overline{t} )^{2}\right) \widehat{\mathbf{D}}_{2}^{-1}\widehat{\mathbf{\Omega} }_{\mathbf{\epsilon}}\widehat{\mathbf{D}}_{2}^{-1}. \] The middle term of $\widehat{\mathbf{V}}_{IV}$ is an estimator of $\mathbf{\Omega}_{\mathbf{\epsilon}}$ given by \[ \widehat{\mathbf{\Omega}}_{\mathbf{\epsilon}}=\widehat{\mathbf{\Gamma} }_{\mathbf{\epsilon}0}+ {\displaystyle\sum\limits_{j=1}^{T-1}} k\left( \frac{j}{M}\right) \left( \widehat{\mathbf{\Gamma}} _{\mathbf{\epsilon}j}+\widehat{\mathbf{\Gamma}}_{\mathbf{\epsilon}j}^{\prime }\right) \text{, \ \ }\widehat{\mathbf{\Gamma}}_{\mathbf{\epsilon}j}=T^{-1} {\displaystyle\sum_{t=j+1}^{T}} \widehat{\scalebox{1.5}{$\bm{\epsilon}$}}_{\theta t} \widehat{\scalebox{1.5}{$\bm{\epsilon}$}}_{\theta t-j}^{\prime}, \] where $\widehat{\scalebox{1.5}{$\bm{\epsilon}$}}_{\theta t}$ is the vector of IV residuals given by \[ \widehat{\scalebox{1.5}{$\bm{\epsilon}$}}_{\theta t}=\left[ \widehat{\epsilon }_{\theta t}^{(1)},\widehat{\epsilon}_{\theta t}^{(2)},\ldots ,\widehat{\epsilon}_{t}^{(n)}\right] ^{\prime}, \] with \[ \widehat{\epsilon}_{t}^{(i)}=y_{1t}^{(i)}-\overline{y}_{1}^{(i)} -\widehat{\theta}^{(i)}\left( y_{2t}^{(i)}-\overline{y}_{2}^{(i)}\right) . \] The long run variance estimator, $\widehat{\mathbf{\Omega}}_{\mathbf{\epsilon }}$, is of the well known kernel form where $k(x)$ is the kernel (downweighting function) and $M$ is the bandwidth tuning parameter that controls the extent of downweighting by the kernel. For the case where one restriction is being tested, $q=1$, a $t$-statistic can be used to test one-sided hypotheses: \[ t_{IV}=\frac{\mathbf{R}\widehat{\mathbf{\theta}}-\mathbf{r}}{\sqrt {\mathbf{R}\widehat{\mathbf{V}}_{IV}\mathbf{R}^{\prime}}}. \]
In order to understand the power properties of $Wald_{IV}$ and $t_{IV}$, their asymptotic limits are derived for local alternatives, $H_{1L}$, of the form \[ H_{1L}:\mathbf{R\theta=r+}\overline{\mathbf{\Delta}}T^{-3/2+\kappa}\mathbf{,} \] where $\kappa$ is the same parameter used in Theorem 1 to model the trend slopes as local to zero. In deriving the asymptotic results, the bandwidth parameter for $\widehat{\mathbf{\Omega}}_{\mathbf{\epsilon}}$ is assumed to be a fixed proportion of the sample size, $b\in(0,1]$, i.e. $M=bT$. Modeling $M/T$ as a fixed constant gives the fixed-$b$ (or fixed-smoothing) limit of $\widehat{\mathbf{\Omega}}_{\mathbf{\epsilon}}$ (and $t_{IV}$). The advantage of the fixed-$b$ approach is that it delivers an asymptotic random variable and associated critical values that depend on the bandwidth and kernel. This is in contrast to appealing to a consistency result for $\widehat{\mathbf{\Omega}}_{\mathbf{\epsilon}}$ which would not depend on the bandwidth or kernel. For more details on the fixed-$b$ approach see bunzel-vogelsang, bunzel-vogelsang, jansson-kvb, kvnewasym, phillips-sun-jin-higherpower, zhang2013fixed, sun-twostepgmm, LLSW18, and LLS21.
Because the form of the fixed-$b$ limit of the test statistics depends on the type of kernel function used to compute $\widehat{\mathbf{\Omega} }_{\mathbf{\epsilon}}$, definitions from kvnewasym are used. A kernel is labelled Type 1 if $k\left( x\right) $ is twice continuously differentiable everywhere and as a Type 2 kernel if $k\left( x\right) $ is continuous, $k\left( x\right) =0$ for $\left\vert x\right\vert \geq1$ and $k\left( x\right) $ is twice continuously differentiable everywhere except at $\left\vert x\right\vert =1.$ The Bartlett kernel (which is neither Type 1 or 2) is considered separately.
The fixed-$b$ limiting distributions are expressed in terms of the following stochastic functions. Let $\mathbf{Q}(r)$ be a generic vector stochastic process. Define the random variable $\mathbf{P}_{b}(\mathbf{Q}(r))$ as \[ \mathbf{P}_{b}(\mathbf{Q}(r))=\left\{
\right. \] where $k^{\ast}(x)=k\left( \frac{x}{b}\right) $ and $k_{-}^{\ast^{\prime}}$ is the first derivative of $k^{\ast}$ from below (left).
The limit of $t_{IV}$ given by Case 1 of Theorem 2 is of the same form as the limit derived by vogelsang-nawaz-JTSA for case of a single trend ratio ($n=1$). Under the null hypothesis (when $\overline{\mathbf{\Delta} }=\mathbf{0}$), the limits of $Wald_{IV}$ and $t_{IV}$ are identical to limits obtained by bunzel-vogelsang in deterministic trend regression models. While the limiting random variables are nonstandard because of the fixed-$b$ limit of $\widehat{\mathbf{\Omega}}_{\mathbf{\epsilon}}$, the $\mathbf{P}_{b}(\widetilde{\mathbf{W}}_{\mathbf{\epsilon}}^{\ast}(r))$ term in the denominator, critical values are available using formulas from bunzel-vogelsang. The critical values depend on the bandwidth through $b=M/T$ and the kernel, $k(x)$.
When the trend slopes are very small or zero, the limiting distributions change, become more complicated, and depend on nuisance parameters. This is to be expected given the weak instrument problem that occurs when the $\beta _{2}^{(i)}$ slopes are small and the fact that $\mathbf{\theta}$ is not defined when slopes are zero. The finite sample simulations will illustrate the extent to which inference breaks down as the trend slopes become very small or zero.
When testing a simple hypothesis about a single trend slope ratio, vogelsang-nawaz-JTSA used Fieller's method (fieller1954) to construct confidence intervals that are robust to very small trends slopes including the case of zero trend slopes. This approach is based on rewriting a simple hypothesis about $\theta^{(i)}$ in terms of a linear restriction involving $\beta_{1}^{(i)}$ and $\beta_{2}^{(i)}$. However, once a null hypothesis involves a linear combination of at least two $\theta^{(i)}$, Fieller's method cannot be applied.
One important empirical application is testing equality of trend slope ratios for two pairs of series. See VMCS26 for tests of equal trend ratios between observed and model generated temperature series. For the null hypothesis of equal trend ratios, it is possible to develop a testing approach similar in spirit to Fieller's method that gives potentially more robust inference when slopes are very small. Suppose there are two pairs of series with trend ratios $\theta^{(1)}$ and $\theta^{(2)}$ and the hypothesis of interest is \[ H_{0}:\theta^{(1)}=\theta^{(2)}, \] or equivalently
Anticipating a local asymptotic calculation, suppose the alternative is specified as local to zero \[ H_{1L}:\theta^{(1)}-\theta^{(2)}=\overline{\mathbf{\Delta}}T^{-3/2+\kappa}. \] Rewriting the alternative hypothesis in terms of the trend slopes gives \[ H_{1L}:\frac{\beta_{1}^{(1)}}{\beta_{2}^{(1)}}-\frac{\beta_{1}^{(2)}} {\beta_{2}^{(2)}}=\overline{\mathbf{\Delta}}T^{-3/2+\kappa}. \] Multiplying both sides of $H_{1L}$ by $\beta_{2}^{(1)}\beta_{2}^{(2)}$ gives
Testing ((ref)) is equivalent to testing
The advantage of using ((ref)) is that the trend slopes can be directly estimated by OLS and ratios are avoided.
To develop a test statistic for testing ((ref)) against ((ref)) define \[ g_{\beta}=\beta_{2}^{(2)}\beta_{1}^{(1)}-\beta_{2}^{(1)}\beta_{1}^{(2)}. \] The natural estimator of $g_{\beta}$ is given by \[ g_{\widehat{\beta}}=\widehat{\beta}_{2}^{(2)}\widehat{\beta}_{1} ^{(1)}-\widehat{\beta}_{2}^{(1)}\widehat{\beta}_{1}^{(2)}, \] where $\widehat{\beta}_{1}^{(i)}$ and $\widehat{\beta}_{2}^{(i)}$ are the OLS estimators ((ref)) and ((ref)).
Because $g_{\widehat{\beta}}$ is a nonlinear function of the slope estimators, its asymptotic variance depends on $\mathbf{\Omega}_{\mathbf{u}}$, the long run variance of $\mathbf{U}_{t}=\left[ u_{1t}^{(1)},u_{1t}^{(2)},u_{2t} ^{(1)},u_{2t\mathbf{U}_{t}}^{(2)}\right] ^{\prime}$, and the vector \[ \mathbf{R}_{\beta}=\left[ \beta_{2}^{(2)},-\beta_{2}^{(1)},-\beta_{1} ^{(2)},\beta_{1}^{(1)}\right] , \] which can be derived using the delta method or, as in the appendix, directly. The feasible version of $\mathbf{R}_{\beta}$ is given by \[ \mathbf{R}_{\widehat{\beta}}=\left[ \widehat{\beta}_{2}^{(2)},-\widehat{\beta }_{2}^{(1)},-\widehat{\beta}_{1}^{(2)},\widehat{\beta}_{1}^{(1)}\right] . \] Let $\widehat{\mathbf{U}}_{t}=\left[ \widehat{u}_{1t}^{(1)},\widehat{u} _{1t}^{(2)},\widehat{u}_{2t}^{(1)},\widehat{u}_{2t}^{(2)}\right] ^{\prime}$ where $\widehat{u}_{1t}^{(i)},\widehat{u}_{2t}^{(i)}$ are the residuals from ((ref)) and ((ref)) estimated by OLS. Define the long run variance estimator of $\mathbf{\Omega}_{\mathbf{u}}$ as \[ \widehat{\mathbf{\Omega}}_{\mathbf{u}}=\widehat{\mathbf{\Gamma}}_{\mathbf{u} 0}+ {\displaystyle\sum\limits_{j=1}^{T-1}} k\left( \frac{j}{M}\right) (\widehat{\mathbf{\Gamma}}_{\mathbf{u} j}+\widehat{\mathbf{\Gamma}}_{\mathbf{u}j}^{\prime}),\text{ \ \ \ } \widehat{\mathbf{\Gamma}}_{\mathbf{u}j}=T^{-1} {\displaystyle\sum\limits_{t=j+1}^{T}} \widehat{\mathbf{U}}_{t}\widehat{\mathbf{U}}_{t-j}^{\prime}. \] Using \[ \widehat{\lambda}_{g}^{2}=\mathbf{R}_{\widehat{\beta}}\widehat{\mathbf{\Omega }}_{\mathbf{u}}\mathbf{R}_{\widehat{\beta}}^{\prime}, \] a $t$-statistic for testing ((ref)) can be constructed as \[ t_{prod}=\frac{g_{\widehat{\beta}}}{\sqrt{\widehat{\lambda}_{g}^{2}\left( {\textstyle\sum\nolimits_{t=1}^{T}} (t-\overline{t})^{2}\right) ^{-1}}}. \] The following Theorem gives the limiting distribution of $t_{prod}$ under the local alternative ((ref)).
When trend slopes are large to small, the asymptotic distribution of $t_{prod}$ has the same form as $t_{IV}$. The two asymptotic distributions are the same under the null ($\overline{\mathbf{\Delta}}=0$) but are different under the local alternative because the variance parameters in the limit are different. As is the case for $t_{IV}$, the asymptotic distribution of $t_{prod}$ changes when trends slopes are very small or zero. Given the complexity of the limits in Cases 2 and 3, it is not clear what Theorem 3 predicts about the finite sample behavior of $t_{prod}$ as trend slopes become very small or zero. Finite simulations in the next section will be used to explore the implications of very small or zero trend slopes.
This section provides simulation results to show some finite sample properties of tests of equal trend slopes. The following DGP was used. The $y_{1t}^{(i)}$ and $y_{2t}^{(i)}$ variables where generated by models ((ref)) and ((ref)) where the noise is given by
\[ \left[
\right] \sim iidN\left( \mathbf{0},\left[
\right] \right) , \] \[ u_{10}^{(1)}=u_{20}^{(1)}=u_{10}^{(2)}=u_{20}^{(2)}=0. \] All the noise series are generated by AR(1) processes. Pairs of series are uncorrelated with each other but there can be within-pair correlation ($\varphi\neq0$).
Given that $t_{IV}$ and $t_{prod}$ are exactly invariant to the values of the intercept parameters, without loss of generality the intercept parameters are set to zero: $\mu_{1}^{(1)}=0,\mu_{2}^{(1)}=0,\mu_{1}^{(2)}=0,\mu_{2}^{(2)} =0$. Results are reported for various magnitudes of $\beta_{1}^{(i)}$ and $\beta_{2}^{(i)}$ where $\theta^{(i)}=\beta_{1}^{(i)}/\beta_{2}^{(i)}=1$ under the null hypothesis of equal ratios across the two pairs. Empirical null rejections are given for $T=50,100,200$ with $10,000$ replications used in all cases. Empirical power is given for $T=100$ where $\theta^{(2)}$ takes on values different from $\theta^{(1)}=1$.
Table 1 reports null rejection probabilities for 5% nominal level tests for testing $H_{0}:\theta^{(1)}=\theta^{(2)}$ against the two-sided alternative $H_{1}:\theta^{(1)}\neq\theta^{(2)}$. Results are reported for the values of $\beta_{1}^{(i)}=\beta_{2}^{(i)}=10,2,.2,.05,0.25,.005,0$ giving $\theta ^{(i)}=1$ except for the case of $\beta_{1}^{(i)}=\beta_{2}^{(i)}=0$ where $\theta^{(i)}$ is not defined. The variance estimators use the Daniell kernel. Results for four bandwidth sample size ratios are provided: $b=A91,0.25,0.5,1.0$, where $A91$ is the AR(1) plug-in data dependent bandwidth proposed by andrews-91. Empirical rejections are computed using fixed-$b$ asymptotic critical values with the critical value function \[ cv_{0.025}(b)=1.9659+4.0603b+11.6626b^{2}+34.8269b^{3}-13.9506b^{4} +3.2669b^{5}\text{,} \] as given by bunzel-vogelsang for the Daniell kernel for $k(x)$.
The top panels of Table 1 give results for the case of iid noise as a benchmark. Here $\phi_{1}^{(1)}=\phi_{2}^{(1)}=\phi_{1}^{(2)}=\phi_{2} ^{(2)}=\varphi=0$. As long as the trend slopes are large, empirical rejections are close to the nominal level of 0.05 for both $t_{IV}$ and $t_{prod}$ and all values of $b$. As the trend slopes get smaller, both tests have rejections below 0.05 (conservative tests). This happens more quickly with $t_{IV}$ than $t_{prod}$ and more quickly with smaller values of $b$ (smaller bandwidths). As $T$ increases, empirical rejections are close to the nominal level for relatively smaller trend slopes. This makes sense given the asymptotic results. For given values of $\beta_{1}^{(i)}$ and $\beta_{2}^{(i)}$, the values of the local trend parameters, $\overline{\beta}_{1}^{(i)}$ and $\overline{\beta}_{2}^{(i)}$, are larger for bigger values of $T$.
The bottom panels of Table 1 give results for serially correlated noise with $\phi_{1}^{(1)}=0.3$, $\phi_{2}^{(1)}=0.7$, $\phi_{1}^{(2)}=0.5$, $\phi _{2}^{(2)}=0.9$. Pairs of series have within-pairs correlation given by $\varphi=0.5$. The first pair of noise series has modest serial correlation in the numerator series, $u_{1t}^{(1)}$, and moderate serial correlation in the denominator series, $u_{2t}^{(1)}$. The second pair of noise series has stronger serial correlation than the first pair with the denominator series again having stronger serial correlation than the numerator series. The relative strengths of serial correlation within pairs are a feature of temperature series where upper atmosphere temperature (numerators) tend to have less serial correlation than surface temperatures (denominators). The relative strength across pairs are a feature of observed versus model generated temperatures where observed temperatures (first pair) tend to have less serial correlation than model generated series (second pair).
As in the iid case, empirical rejections in the serial correlation case tend to fall as trend slopes become smaller indicating that the tests remain conservative when trend slope are small (or even zero). For larger trend slopes, the positive autocorrelation can lead to over-rejections especially if the data-dependent bandwidth is used. As $b$ increases, the tendency to over-reject is mitigated. This is a well-known property (see bunzel-vogelsang and kvnewasym among others). It is not surprising that over-rejections are larger with the data-dependent bandwidth because those bandwidths tend to give relatively small values of $b$. Comparing $t_{IV}$ and $t_{prod}$, one can see that $t_{IV}$ tends to over-reject less than $t_{prod}$ especially when using the data-dependent bandwidth. In contrast, $t_{IV}$ tends to under-reject more than $t_{prod}$ as trend slopes become smaller. The main takeaway is that $t_{IV}$ over-rejects less than $t_{prod}$ when bandwidths are small and trend slopes are large, but $t_{IV}$ is more conservative when trend slopes are small (regardless of bandwidth). As a practical matter, using the slightly larger bandwidth of $b=0.25$ gives non-trivial reductions in over-rejections relative to the $A91$ bandwidth rule.
Table 2 gives results for empirical power of the tests. Results are only given for the case of serially correlated noise (patterns are similar for iid noise) with $T=100$. In all cases $\theta^{(1)}=1$, and results are given for a grid of values of $\theta^{(2)}$ above and below $1$. The range of the grid increases as the trend slopes decrease to provide information about the power curves. It is not surprising that in order to see power, the range of $\theta^{(2)}$ needs to be increased as trend slopes decrease - this is predicted by the slower rate of convergence of $\widehat{\mathbf{\theta}}$ (larger sampling variance) as trend slopes become smaller. Empirical power is not size-adjusted to show actual power in practice. Null rejections are in bold for the cases where $\theta^{(2)}=1$.
When trend slopes are large, power is similar for $t_{IV}$ and $t_{prod}$ for the bandwidths $b=0.25,0.5,1.0$. Power is roughly symmetric around null value of $\theta^{(1)}=1$. When the $A91$ data-dependent bandwidth is used, $t_{prod}$ over-rejects more than $t_{IV}$ and has correspondingly higher power. With smaller trend slopes some differences in power emerge between $t_{IV}$ and $t_{prod}$ that are not solely due to differences in null rejections. For example, when $\beta_{2}^{(1)}=\beta_{2}^{(2)}=0.2$, $t_{IV}$ and $t_{prod}$ have similar null rejections with $b=0.5,1.0$. In these cases power of $t_{IV}$ is higher for $\theta^{(2)}<1$, whereas power of $t_{prod}$ is higher for $\theta^{(2)}>1$. Power for both tests is higher with smaller values of $b$ --- another well-known feature of tests based on kernel variance estimators that use fixed-$b$ critical values. As trend slopes become smaller, power of both tests decreases and power can be low even for values of $\theta^{(2)}$ very far from $1$. Again, this is not surprising because smaller trend slopes have relatively less information about $\theta^{(1)}$ and $\theta^{(2)}$ for given strength of the noise. Interestingly, with very small trend slopes, $\beta_{2}^{(1)}=\beta_{2}^{(2)}=0.05,0.025$, power initially increases as $\theta^{(2)}$ moves away from the null value of $1$ but can begin to fall when $\theta^{(2)}$ is very far from $1$.
The main takeaways from the power results are: i) for large trend slopes, $t_{IV}$ and $t_{prod}$ have similar power, ii) for medium to small trend slopes power cannot be ranked, iii) power of both tests decreases as the bandwidth increases and iv) there is low power in detecting differences between trend ratios when trend slopes are small.
The theory and simulations indicate that $t_{IV}$ and $t_{prod}$ perform similarly in practice when i) trend slopes are not very small, and ii) small bandwidths are avoided when there is nontrivial serial correlation in the noise. Both tests can over-reject when there is positive serial correlation in the noise although this problem is mitigated by larger sample sizes. For very small trend slopes both tests become conservative with $t_{IV}$ becoming conservative more quickly than $t_{prod}$. Power of the tests cannot be ranked in this case. Because of the conservative nature of both tests when trend slopes are small (or even zero), any rejections obtained for tests of equal ratios are robust to very small trend slopes. The price paid for this robustness is lower power, but one cannot expect trend ratios to be precisely estimated when trend slopes are very small.
If a practitioner uses either test (or both) with the Daniell kernel and a non-small bandwidth for the variance estimator, then rejections of equal ratios can be viewed as relatively robust to stationary serial correlation in the noise and very small trend slopes.
VMCS26 used the methods developed in this paper to test equivalence between trend ratios of observed temperature series and temperature series generated by recent runs of climate models. They compared trend ratios of temperature series at various atmospheric heights relative to trends in surface temperatures for the tropics region of the earth. These trend slope ratios are called amplification ratios because climate models predict amplified warming in the lower to mid troposphere relative to the surface in the tropics. VMCS26 found that climate models tend to have more amplification that seen in observed temperature series and that the differences are statistically significant using the $t_{IV}$ and $t_{prod}$ test statistics. While VMCS26 compared amplification ratios between each of five sets of observed temperatures with each of 39 sets of climate model temperature series, they did not compare amplification ratios among the five sets of observed temperature series. Comparisons across observed series is interesting given the different methods by which the observed series are measured, constructed, and aggregated.
VMCS26 provide details on the five sets of observed temperature series which can be summarized as follows. Observed temperatures for various pressure levels in the atmosphere are measured in two ways in the five sets of observed temperatures. The first method uses station-based, balloon-borne radiosonde records. Temperature data is collected at specific pressure levels, generally up to 20 hectopascals (hPa) which is approximately 27 km above the earth's surface. A smaller hPa value indicates a higher level about the surface. The second method is known as reanalyses of weather data combined with climate models to generate temperature series. The five observed data series include three from balloons. Two are data series from the University of Vienna, RAOBCORE v1.9 and RICH v1.9. The third is from the U.S. National Oceanic and Atmospheric Administration, the RATPAC-A v2 data series. These sets of data are labeled RAOB, RICH, and RATP. The two global reanalyses data sets are taken from the European Centre for Medium-Range Forecasts Reanalyses (ERA5), and the Japanese Reanalyses for Three Quarters of a Century (JRA3Q).
The data is aggregated on an annual basis and spans the years 1958 to 2024 (67 years) for the surface and the grid of hPa levels 850, 700, 500, 400, 300, 200, 150, 100, 70, 50, 30, 20. Data is not available for the RATP data set for hPa 20. Empirical results are presented in three tables where in all cases confidence intervals at the 95% level are computed using the same variance estimators and fixed-$b$ critical values used in the finite sample simulations (Daniell kernel, A91 bandwidth, fixed-$b$ critical values).
Table 3 provides summary statistics in the form of OLS estimated linear trend slope parameters using regression ((ref)) for each of the five sets of temperatures across the hPa levels. Estimated trend slopes and confidence intervals are scaled to be in units of degrees Celsius per decade. Nearly all of the estimated trend slopes are statistically significantly different from zero. For each of the five sets of series, there is a clear pattern of trend slopes across hPa levels. There is warming at the surface. There is less warming at 850 hPa but more warming from hPa levels 700 to 150. At hPa 100 up to 20, there is cooling with cooling increasing at higher altitudes.
Table 4 reports estimated trend ratios for each hPa level relative to the surface. Confidence intervals are computed using Fieller's method following vogelsang-nawaz-JTSA. All five sets of observed temperatures show the same pattern in estimated trend ratios across hPa levels. Amplification (ratios greater than 1) occurs for hPa levels 700 to 200. For hPa level 150 two observed series show amplification whereas three do not. For hPa levels 100 to 20, there is cooling at these higher altitudes and ratios are negative and increase in magnitude for higher altitudes (lower hPa values). While the patterns across hPa levels are similar for the five sets of observed temperature series, there can be noticeable variation across the five sets for a given hPa level. This raises the question as to whether ratios are equal across pairs of observed series for a given hPa level.
Table 5 reports, for each hPa level, pairwise differences between estimated ratios, $\widehat{\Delta}_{\theta}$, and pairwise $g_{\widehat{\beta}}$ values. 95% fixed-$b$ critical values are given below each estimate. A $^{\ast}$ superscript on an estimated value indicates a rejection of the null hypothesis of equal ratios (indicates the confidence interval does not contain the value $0$). For reporting purposes, the values of $g_{\widehat{\beta}}$ and confidence intervals are scaled by $10^{4}$. Of course, this has no effect on whether the null hypothesis of equal ratios can be rejected. Table 5 is divided into four panels where each panel reports results for three hPa levels. In general, there are many pairs where the differences in trend ratios are not small in magnitude and are statistically significant. For example, among the three balloon data sets, RICH, RAOB, RATP, there are rejections of equal ratios for 8 to 11 of the hPa levels across the three respective pairs. One case where there is relative alignment of trend ratios is between the two Reanalyses data sets, ERA5 and JRA3Q, where rejections of equal ratios occurs for only 3 of 12 hPa levels. Except between the pair of Reanalyses data sets, the results in Table 5 suggest there are differences in trend ratios among pairs of the observed data sets across hPa levels.
This paper develops estimation and inference methods for systems of pairs of trend stationary time series where the parameters of interest are ratios of trend slopes between pairs of time series. Inference focuses on null hypotheses that can be written as linear restrictions across trend ratios. Trend slopes are estimated by IV using time as an instrument and test statistics are robust to i) stationary serial correlation in the fluctuations around trend, and ii) correlation between and across pairs of time series. For the empirically relevant special case of testing for equal trend slopes between two pairs of time series, an alternative testing approach is developed by restating the equal trend ratio restriction as a restriction involving products of the underlying trend slopes which are estimated by OLS. Theory and finite sample results suggest that both the IV and products approach work well and have similar properties when trend slopes are not too small and serial correlation is not too strong. When trend slopes are very small relative to the variation around the trend function, the null limiting distribution of both tests become nonpivotal and both tests tend to under-reject under the null and have low power. Lower power is expected when trend slopes are very small because trend ratios cannot be precisely estimated. That both tests become conservative when trend slopes are very small gives the tests useful robustness to very small trend slopes. Overall, the IV and product approaches have complementary finite sample properties and using both is recommended in practice for testing the hypothesis of equal ratios across two pairs of time series.