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-0.01cm darkpowderblue A Neural Phillips Curve and a Deep Output Gap .
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Few equations are as central to modern macroeconomics and current monetary policy debates as the Phillips Curve (PC) -- and its modern incarnation, the New Keynesian Phillips Curve (NKPC). Yet, many problems plague its estimation and thus, our understanding of how increasing economic activity translates into higher pressures on the price level suffers. Similarly, our understanding of how inflation expectations influence current inflation is also compromised.
This paper focuses on a predictive Phillips curve -- building an equation that uses, among other things, some measures of real activity and expectations to forecast inflation. It provides a new solution to a pervasive problem in empirical inflation modeling and economics research in general. Namely, the two key components of the NKPC, inflation expectations ($\mathcal{E}_t$) and the output gap ($g_t$), are both unobserved. Instantly, this opens the gates to a zoo of proxies. Which gap to choose? Which inflation expectations at what horizon, and from whom? Those are crucial empirical choices on which theory is mostly silent. $\mathcal{E}_t$ and $g_t$ are necessary to produce and understand inflation forecasts, both of which are needed to guide monetary policy action.
{ \sc A {Hemisphere} Neural Network.} Taking a step back, what basic macroeconomic theory suggests is that two sufficient statistics summarizing different groups of economic indicators should predict inflation reasonably well. More precisely, we know that (i) there should exist some abstract output gap, or in other words, a possibly nonlinear combination of variables related to the state of the economy (labor markets, industrial production, national accounts) that influence inflation, and (ii) some combination of price variables (past CPI values and several others) and other measures of inflation expectations also impact inflation directly. I make this vision operational by developing a new Deep Neural Network (DNN) architecture coined Hemisphere Neural Network (HNN). The DNN is restricted so that its final inflation prediction is the sum of components composed from groups of predictors. These group of predictors are separated at the entrance of the network into different hemispheres, defined as modules or sub-networks operating in isolation until their outputs are combined in a final layer. This particular structure allows the interpretation of the top layer's cells output as key macroeconomic latent states in a linear equation -- a Phillips curve. Moreover, the estimation of time-varying PC coefficients and the key latent states can be performed within a single model.
While HNN's development is motivated from inflation predictive modeling, its applicability extends to the various problems in economics where the link between theoretical variables and actual variables in our databases is not crystal clear. Examples include the neutral interest rate, Taylor rules inputs, term premium, and of great interest recently, "financial conditions" adrian2019. This also extends to poorly measured observed explanatory variables. From an econometric perspective, this paper introduces a novel approach, leveraging modern deep learning to address mismeasurement issues impeding empirical macroeconometrics. Obviously, HNN is by no means the first methodology dealing with latent state extraction or attenuation bias. But, when compared to the older generation of methods, its empirical merits will be decisive. As such, this paper sits at the intersection of three literatures: (i) estimating the output gap and Phillips curves, (ii) the application of deep learning methods in macroeconomic forecasting, and (iii) interpretable artificial intelligence (AI). I now review those concisely and discuss how HNN may improve upon existing methods.
{\sc Output Gap and Phillips Curves.} By the virtues of being a supervised learning problem, HNN improves over methods where $g_t$ is extracted from an economic activity series and then, optionally, evaluated in a "second stage" PC regression. That is, ${g}_t^{\text{HNN}}$ is by construction the most relevant summary statistic of real activity to explain inflation. The unsupervised approach -- filtering either GDP, unemployment, or related indicators in a certain way -- is by far the most widespread practice in central banks. There are now alternative filtering tools hamilton2018you, and others that incorporates additional (timely) information berger2020nowcasting,de2017real. Taking a step back, there is the deeper question of whether this filtered $g_t$ is what we should be after at all, especially that its explanatory power for inflation seems to be vanishing quickly blanchard2015inflation. Little prevents the true $g_t$ from falling outside the range of what trend-cycle decompositions can extract. HNN surely has its own set of assumptions -- like that real activity should impact inflation through a linear reduced-form equation -- but I will argue throughout that they are more palatable.
With respect to econometric methods that included some mild form of supervision in the estimation of $g_t$ kichian1999measuring,blanchard2015inflation,chan2016,chan2018,hasenzagl2018model,jarocinski2018inflation, HNN improves by dispensing with restrictive law of motion assumptions inherent to a state-space methodology. This is not innocuous, as the smoothness and composition assumptions plausibly drive results to an appreciable extent, leading us back, at least in part, towards unsupervised methods. Additionally, with respect to the aforementioned approaches, HNN easily handles a high-dimensional group of inputs for both $g_t$ and $\mathcal{E}_t$ and performs computations quickly using standard highly optimized deep learning software. Nonlinearities in how activity variables translate into $g_t$ or $\mathcal{E}_t$ (and ultimately $\pi_t$) are allowed through a deep and wide network architecture with over 2 million parameters. In that sense, HNN fully embraces the implications from the widely documented {double descent} phenomenon belkin2019reconciling by being overtly overparametrized and yet providing good non-overfitting results. Nonlinearities are in fact a necessary feature given the accumulating evidence that the PC might be nonlinear with respect to traditional slack indicators linde2019resolving,MRFjae,forbes2021low,benigno2023s.
{\sc Macroeconomic Forecasting and Interpretable AI.} Regarding applications of modern neural networks to inflation forecasting (such as hauzenberger2020real and paranhos2021predicting), HNN is an inherently interpretable model, tailored to inflation by incorporating minimal "theoretical" restrictions. These restrictions enable the outputs of the last layer to be understood as economic states. In its simplest form, HNN is a generalized additive model hastie2017generalized where more than one regressor is allowed to enter each linearly separated nonparametric function, and all functions are learned simultaneously through a gradient-based approach. By that, HNN fits within what hothorn2010model defines as structure-based additive models or group-level structured neural networks in bianchi2021. Closely related, agarwal2020neural and others develop architectures inspired from generalized additive models to enhance interpretability in deep networks for generic tasks. While these articles certainly tackle some of the opacity issues coming from fully nonparametric estimation, none address those that are inherent to non-sparse high-dimensional (even linear) regression. This also represents a bottleneck for post-hoc interpretability methods, such as Shapley values, which typically require some form of input pre-selection to yield economically meaningful results buckmann2022interpretable. The key observation behind HNN's design is that grouping variables into hemispheres and combining their outputs according to theory enables the interpretation of the high-dimensional nonlinear black box as a sparse linear unobserved components model. Thus, more generally, HNN is related to interpretable AI approaches aiming for user-centered explanations, addressing how AI systems outputs can be communicated effectively to (human) decision makers Gunning2019,Miller2019.
{\sc Results.} Two main variants of HNN are proposed. They primarily differ in their handling of time-variation in the model. The first one (HNN) is less restrictive on how exogenous time-variation mixes with other nonlinearities. The caveat is that only the gap's contribution to inflation can be extracted from the model. The second architecture (HNN-F, the flagship model) has a built-in factorization which disentangles $g_t$'s estimates from its exogenously time-varying coefficient.
Many new insights are obtained. First, forecasts are better than traditional PC-based forecasts. For the post-2008 period, this can be partly attributed to HNN-F's gap -- projected out-of-sample -- closing much faster than traditional ones, then slipping back gently into negative territory in the mid-2010s. HNN also captures the 2021 upswing in inflation and attributes it to a strongly positive output gap. While both peaks are comparable in size to the 1970s, the components show much less persistence than they did four decades ago---in line with the stop-and-go nature of economic constraints of the pandemic era. Second, throughout the whole sample and for both architectures, the contribution of the output gap component is shown to be significantly higher than what is reported from time-varying PC regressions with traditional gap measures. Conversely, the effect of the expectations component is found to be milder overall, with the notable exception of 2021, where it moves upward quickly while traditional estimates remain flat. Third, the Phillips curve coefficient in HNN-F decreased in the early 1980s and experienced a revival starting from the 2000s. This contrasts with traditional PC regressions, which suggest a very weak relationship in recent years. As a result, HNN-F -- through its positive gap and alive-and-well PC coefficient -- forecasts the inflation awakening of 2021. Fourth, important input variables driving $g_t$ are found to be hours worked and vacancies. While the former points towards the importance of the intensive margins, giving indication why an unemployment-based gap might underperform, the prominent role of vacancies highlights that a better leading indicator appears to be the demand side rather than the supply side of the labor market. A surrogate model analysis shows that the shape of the linkage between those two indicators and $g_t$ is highly nonlinear -- reminiscent for vacancies of the hockey stick shape in benigno2023s.
{\sc Outline, Supplementary Material, and Extensions.} Section (ref) introduces HNN, motivates its structure, and discusses practical aspects. Section (ref) conducts the empirical analysis and Section (ref) presents forecasting results. Section (ref) concludes. Supplementary material includes a simulation study as well as many robustness checks and variations on the main specification. It also contain vignettes highlighting the wide applicability of the method. The first extension leverages sims2019four's 4-equation NK model to extract a more focused indicator of latent "credit conditions". The second creates a supervised composite from a panel of international GDP growth data to evaluate the effect of global economic conditions on US inflation. Those extensions show the wide applicability of the method when theory implies testing with empirically ambiguous indicators. Lastly, a more significant extension is considered in follow-up work: GCFK obtain and evaluate density forecasts by introducing an additional "volatility" hemisphere and altering the loss function used to train the model.
This section discusses the motivation behind the newly proposed network architecture. Basic New Keynesian theory tells us that current inflation ($\pi_{t}$) is the sum of the contributions of inflation expectations for the next period ($E_t(\pi_{t+1})$) and the output gap ($g_t$). stock2008phillips define Phillips curve forecasts as those which are structured so to feature (at least) both of these components. Defining expectations less stringently as $\mathcal{E}_t$, the simple $s$-steps ahead predictive problem would be formulated as
where $\nu_{t+s}$ is unpredictable noise. Before even thinking about the definition of $\mathcal{E}_t $ and $g_t $, (ref) needs to be refined to include two features that are known to matter when bringing this equation to the data. First, coefficients may evolve through time ($\theta_t$ and $\gamma_t$), which has become an empirical necessity to accurately describe inflation in most advanced economies blanchard2015inflation. Second, commodity price cycles (particularly energy) play an important role in driving short-run inflation fluctuations, having a direct impact on it (i.e., not channeled through the real activity component). In fact, the typical worry is that omitting it, using hasenzagl2018model's words, may "overpower" the Phillips curve. While this is certainly more of a worry when setting the modeling problem as a contemporaneous one, we would not want the omission to lead to, e.g., an overstatement of the importance of real economic activity. The equation featuring all those empirically-desirable features is
Essentially, this is a 3-factor model where $h_{t,\mathcal{E}}$, $h_{t,g}$, and $h_{t,c}$ are contributions. Thus, let $\mathcal{H}_g$, $\mathcal{H}_\mathcal{E}$, and $\mathcal{H}_c$ be the expectations, real activity, and commodities prices hemispheres, respectively.\footnote{The terms "gap" and "output gap" are used throughout the paper in a loose fashion, meaning they refer to a generic latent indicator of economic non-slack. That is, it refers to an abstract gap between aggregate demand and aggregate supply conditions, not a deviation from the trend of a particular observed measure of economic output. That being said, there is no claim that $g_t$ is the "true" wedge between aggregate demand and aggregate supply, but it is a useful indicator that is optimized to match the implications we would hope such a true $g_t$ to have.} To make this operational, I impose some restrictions on a fully connected NN so that its $h_t$'s will carry economic meaning. A shallow and narrow (for visual convenience) HNN architecture for the three hemispheres case is displayed in Figure (ref).
Some remarks are in order. First, HNN's architecture is trivially extendable to more than 3 hemispheres.\footnote{ I use the term hemisphere throughout, but modules or sub-network could be equally adequate. The appellation "hemisphere" is very loosely inspired from neurons in the AI model being allocated to different parts of a brain assembling outputs to produce a final decision, although here it is detached from its etymological sense of being two parts of a sphere.} This makes it convenient for splitting some hemispheres into sub-hemispheres (like expectations into short-run vs. long-run). It also makes it a flexible testing ground for theories claiming the NKPC should be augmented with something, but that something is not clearly defined in terms of what is in our databases. Such extensions are considered in Appendix \phantomsection{(ref)}.
Second, HNN does not give us $g_t$ nor $\gamma_t$, but their product $h_{t,g}$. This is not the neural network's doing, but rather the design of the problem. With $g_t$ and $\gamma_t$ both unobserved and possibly time-varying, they cannot be separately identified without additional assumptions on how $g_t$ and $\gamma_t$ should or should not evolve through time. HNN-F, developed and motivated in Section (ref), will propose such restrictions and leverage them to separate $g_t$ and $\gamma_t$. The more general point being made is that the plain HNN architecture provides $h_{t,g}$ as the most sensible output given the econometric conditions, but nothing prevents a researcher from splitting it in $g_t$ and $\gamma_t$ using whichever assumptions deemed reasonable. Nonetheless, for policy purposes, a crucial use of $g_t$ is to inform us on how real activity contributes to $\pi_t$ -- and that is the HNN's direct output. Finally, this does not prevent from comparing HNN results with other methodologies since their gap's contribution to inflation can easily be calculated from the PC regression (see Section (ref)).
Third, a comment on the group-wise network structure and the "separability assumption". Precisely, by separability, it is meant that $h_{t,j}$'s are the product of mostly non-overlapping (they share $t$ in common) groups of predictors. Of course, it is possible that the interaction of the prices group and the real activity group influences inflation. On the other hand, some level of separability is crucial for interpretability in this high-dimensional environment. It is the separation, as suggested by the (linearized) PC, that gives $h_{t,j}$ its meaning. While there is nothing sacred about linearized PCs, it is noteworthy that the proposed separation is not new to HNN at all. It is inherent to almost any linear PC estimation (there is a block of lags, and an output gap, all separated and typically non-interacting). As a side note, some overlap between the contents of $\mathcal{H}$'s is absolutely possible if the definition of $h_{t,j}$'s calls for it. Finally, $h_{t,j}$ need not be orthogonal since they are obtained from what is ultimately a (supervised) semi-nonparametric regression which dispenses with most of the traditional identification problems inherent unsupervised learning and factors models.
The baseline estimation is at the quarterly frequency using the dataset FRED-QD mccracken2020fred. It is \href{https://research.stlouisfed.org/wp/more/2020-005}{publicly available} at the Federal Reserve of St-Louis' website and contains 248 US macroeconomic and financial aggregates observed from 1960Q1. The target considered in the main analysis is CPI Inflation (thus $\pi_{t+1} = \Delta log(\text{CPI}_{t+1})$). Forecasting and some robustness checks on $g_t$ are conducted using core inflation ($s=1$) and year-over-year (YoY) headline CPI four quarters ahead ($s=4$). The transformations to induce stationarity for predictors are indicated in mccracken2020fred.
Our empirical baseline model comprises 4 hemispheres. It consists of the 3 described in Section (ref), with one of them being split in two sub-hemispheres. Precisely, to examine them separately, I split expectations into two additive components: long-run/exogenous ($\mathcal{E}_t^{\text{LR}}$), and short-run ($\mathcal{E}_t^{\text{SR}}$). The remaining two $\mathcal{H}$'s are real activity and commodity/energy prices. For each variable $X_{t,k}$, I include 4 lags of it and 3 moving averages of order 2, 4, and 8. This is motivated by MDTM's so-called Moving Average Rotation of X (MARX) transformation, developed to alter the implicit prior of certain machine learning (ML) algorithms when applied to time series data -- without recoding them. $\mathcal{H}$'s composition details are in Table (ref) and the complete list of FRED-QD mnemonics is available in Appendix (ref).
In addition to FRED-QD's price series, I include inflation expectations/forecasts from the Survey of Professional Forecasters and the Michigan Survey of consumers. Proxying directly for inflation expectations with survey-based data emerged as a popular alternative to fully rational expectations coibion2015phillips,coibion2018formation. The downside is that theory provides little to no guidance about the provenance of expectations that should be used. meeks2023heterogeneous use a functional principal component approach to summarize the distributional aspect of the expectations from the Michigan survey of consumers (among others) and finds that the additional information reduces the role of inflation persistence. It is worth noting that this line of papers almost universally take the unemployment/output gap as given. This paper, for simplicity and to maximize the length of the historical period being studied, opts for very standard series of inflation expectations as inputs, like the average expectations from professional forecasters and consumers surveyed by the University of Michigan, as well as lags from price indices. From a methodological and practical standpoint, nothing prevents the inclusion of a much richer and heterogeneous set of beliefs -- these would be additional regressors in $\mathcal{H}_{\mathcal{E}^\text{SR}}$. By construction, the HNN procures the optimal "summary statistic" of such expectations because the nonlinear information compression parameters are estimated in a supervised fashion. Thus, in future work, HNN could digest larger expectations information sets (like the whole cross-section dimension of a survey, or many quantiles of it).
{\sc On the Separation of Expectations.} An important question for the interpretation of HNN outputs as plausible estimates of the latent concepts "economic slack" and "inflation expectations" is what is an appropriate composition for the expectations hemisphere, as many series are reasonable candidates to characterize consumers' information set. The motivation from using lags of the target is obviously to proxy for backward-looking expectations in a way that is traditionally done in tightly specified linear Phillips curves. Then, the argument for including many price sub-indices is that consumers typically pay different levels of attention to individual prices to form expectations, and the overall weighting likely differs from headline consumption basket weights (being used implicitly through lags of $\pi_t$). However, this very inclusive set of predictors may be a threat to HNN separability assumptions, necessary for the economic interpretation of results. Appendix (ref) reports, among other things, results where the $\mathcal{H}_{\mathcal{E}^\text{SR}}$ is cleared of anything but survey-based expectations series. We will see that results on real activity remain consistent across alternative expectations specifications whereas results on expectations inevitably differ in certain historical episodes. The data-rich $\mathcal{H}_{\mathcal{E}^\text{SR}}$ in the baseline HNN specification is found to be more timely, with the resulting series rising from its bed faster post-2020 than specifications only using survey-based information.
{\sc Timing Assumptions.} The baseline HNN specification is set up as a one-step-ahead direct forecasting problem, in the spirit of stock2008phillips's Phillips curve inflation forecasts and the widespread use of such equations in central banking to build an inflation outlook yellen2017inflation. Beyond the prism of forecasting, HNN, being a direct forecasting problem, can also be interpreted as the first step of a local projection studying the dynamic impact of real activity and expectations on inflation. This is similar to studies that examine the multi-horizon impacts of real activity shocks, considering that it takes time for real activity conditions to fully materialize in inflation numbers del2020s. The practical merits of the "forecasting" approach are evident, such as directly creating forward-looking metrics useful for predicting inflation, rather than performing a two-step process as in, e.g., banbura2021inflation. However, the timing assumptions embedded in the baseline HNN are not entirely aligned with the macroeconomic theory that motivated it. The NKPC yields a contemporaneous regression ($\pi_t$ regressed on $E_t(\pi_{t+1})$ and $g_t$), as opposed to using $\pi_{t+1}$ as the dependent variable. Results are reported in Appendix (ref), where an HNN specification is built using a more theory-consistent timing of the target variable.
As a consequence of sparing HNN from the numerous assumptions typically associated with output gap extractions, the procedure only produces $h_{t,g}$, the gap's contribution to inflation, rather than $g_t$ itself. It was discussed that splitting $h_{t,g}$ into $g_t$ and $\gamma_t$ can be done if the researcher is willing to use additional assumptions on $g_t$ and $\gamma_t$. Denote $\mathcal{H}_j \setminus k$ as the restricted set where variable $k$ is excluded from the set of predictors originally included in the hemisphere $j$. One possible factorization is $\gamma_t=h_{\gamma}(t)$ and $g_t=h_{g}(\mathcal{H}_g \setminus t)$. The factorization coerces the PC coefficient to move exogenously and slowly -- like what is assumed by random walk coefficients in {\color{PineGreen} Chan, Koop, and Potter (2016)} (henceforth CKP) and many others. This is merely an interpretation device because what we can say about $g_t$ and $\gamma_t$ depends perfectly on what we assume they can be. For instance, a convex PC is ruled out by $\gamma_t=h_{\gamma}(t)$ but residual "convexity" will be mechanically relegated to $g_t$. Nonetheless, what HNN-F provides is a $g_t$ which definition (i.e., its composition out of real activity data) is constant over time. The coefficient is a slow-moving scaling coefficient ($\gamma_t$) -- which can be assumed fixed for short- and medium-run forecasting horizons. In a way, HNN-F tries to build a pair of $g_t$ and $\gamma_t$ which are most convenient from a macroeconomic monitoring perspective: a fixed definition of the gap which passthrough to inflation is very stable. Most of the "complexity" is channeled through the definition of the gap itself.
Implementing $h_{t,g}=h_{\gamma}(t)h_{g}(\mathcal{H}_g \setminus t)$ is easy within HNN and the PyTorch (Python) or Torch (R) environments. First, an additional hemisphere containing only $t$ is created. Then, in the final layer, rather than summing 3 or 4 $h_{t,j}$'s as in Figure (ref), some last layer outputs will be multiplied together. Namely, the output of the hemisphere containing only $t$ will be multiplied with that of $\mathcal{H}_g \setminus t$ and the product will be added to the rest of the sum constituting the neural PC. For consistency, this intuitive factorization is forced on each component. Thus, using the notation established in (ref), the final layer in HNN-F (F for factorized) will be
Clearly, the various $h_t$'s of (ref) are not scale- and sign-identified, except for $ h_{\mathcal{E}^{\text{LR}}}(t) $ since it is not multiplied with any other component. To identify the relevant $h_t$'s, time-varying coefficient hemisphere outputs $\theta_t$, $\gamma_t$ and $\zeta_t$ are all forced to be non-negative by feeding them forward through an absolute value layer before they enter the final layer above. This prevents the gap from being the symmetrical opposite of what it is expected to be. Moreover, the scale identification problem is fixed in estimation by setting the mean (over all $t$'s) of time-varying coefficients to 1. One can rescale gaps and coefficients after estimation. In Section (ref), $g_t$'s standard deviation is set to that of Congressional Budget Office's (CBO) gap to facilitate comparison. By estimating $g_t$ flexibly and allowing for $\gamma_t$ to vary over time, HNN-F allows for an investigation of the declining link between real activity and inflation with a lessened worry that a declining $\gamma_t$ is due to a mismeasured $g_t$. This hypothesis has often been ruled out when using traditional gaps or economic indicators in the context of linear econometric methods stock2019slack,del2020s.
Within each $\mathcal{H}$, we have a standard feed-forward fully connected network. We set $\texttt{layers} = 5$ and $\texttt{neurons} = 400$. For HNN, I maximize efficiency by enabling weight sharing nowlan1992simplifying,bender2020can across hemispheres. In other words, nonlinear processing parameters are forced to be identical across hemispheres. In HNN-F, I relax that constraint and the states hemispheres are given $\texttt{neurons} = 400$ and $\texttt{layers} = 3$ while the coefficients hemispheres (with only input being $t$) have $\texttt{neurons} = 100$ and $\texttt{layers} = 3$. More layers or neurons beyond that point visibly increase what is apparent noise in the components, and not improve out-of-bag MSE.
The maximal number of epochs (optimizer steps) is fixed at 500. The activation functions are all ReLU ($\operatorname{ReLU}(x)=\max \{0, x\}$) and the learning rate is 0.005 for HNN and 0.05 for HNN-F. The adequacy of such values can be assessed by examining optimization curves (along epochs). 65% of the training sample is used to estimate the parameters and the MSE of the remaining 35% is used to determine when to optimally stop optimization -- early stopping being known to perform a form of ridge regularization on network weights raskutti2014early. This random shuffling of data is done through shuffling blocks of 8 quarters for quarterly data. Users only interested in point estimates (and not, e.g., getting bands around $g_t$) can increase the subsampling rate to higher values such as 85%. The batch size is the whole sample and the optimizer is Adam. For forecasting, I do 50 random 65-35 allocations of data and ensemble resulting predictions. This is beneficial in two aspects. First, it stabilizes the optimal early stopping point choice. Second, it is known that ensembling overfitting ("interpolating") networks can give a performance similar to that of very large yet computationally costly networks, by among other things, integrating out noise coming from network weights initialization d2020double. Finally, I perform a mild form of dropout by setting the dropout rate to 0.2. The typical z-score normalization of inputs in neural networks needs to be adjusted to the HNN case and such details can be found in Appendix (ref). The R and Python codes are available on \href{https://github.com/philgoucou/hemisphere}{Github}. They feature the functions used throughout this paper and a replicating example code of the main empirical results.
While the large total number of neurons may seem like an ill-advised choice given the lasting threat of overfitting, there is a now a wide literature documenting the benefits of overparametrization in such models belkin2019reconciling. Moreover, their documented stability in optimization (and with respect to initialization of weights) will prove useful when constructing bands from the ensemble. The simulation study in Appendix (ref) documents that a large HNN architecture is indeed more apt at recovering precisely "true" latent states that are nonlinear functions of inputs.
Ensembling requirements are more demanding to quantify $h_{t,j}$'s uncertainty than to obtain point estimates and forecasts. $B$, the total number of runs, is set to 300 and blocks of 2 years are subsampled (without replacement) with a subsampling rate of 0.65. This takes about an hour to run on an M1 MacBook Air.
Since any DNN can easily fit the training data much better than it actually does on the test data, it is wiser to opt for an out-of-bag strategy in order to calculate $h_{t,j}$'s in-sample as well as their quantiles. Such a strategy was deployed in MRFjae for time-varying coefficients obtained from a Random Forest (RF). Given that HNN also uses dropout to a mild extent and is optimally early-stopped to maximize hold-out sample performance, this additional precaution may not appear necessary at first sight. However, it is the object of a burgeoning literature of its own that best-performing DNNs out-of-sample can very well overfit in-sample belkin2019reconciling. This obviously complicates things for in-sample analysis of historical estimates, and considering out-of-bag estimates is a natural solution to that problem inspired from metrics traditionally reported for RF.
The out-of-bag calculations proceed as follows. Assume we have a sample of size 100. I estimate HNN using data points from 1 to 65, and project it "out-of-bag" on the 35 observations not used in training. This gives us ${h}_{65:100,b}$ for a single allocation $b$ while ${h}_{1:65,b}$ are still NAs. By considering many such random (block) allocations where "bag" and "out-of-bag" roles are interchanged, I obtain the final $h_{t,j}$'s by averaging over $B$ at each $t$ such that
This constitutes an approximation to a Block Bayesian Bootstrap by replacing the posterior tree functional $\mathcal{T}$ in MRFjae by HNN. Thus, ${h}_{t,j,b}$ draws can be used to compute credible regions. This relies on the connection between Bagging and the Bayesian Bootstrap put forward for Random Forest by taddy2015bayesian. More recently, newton2021weighted develop a weighted Bayesian Bootstrap, derive theoretical guarantees, and show its applicability to trend filtering, deep learning, and other problems.
How should we think of the statistical adequacy of HNN and HNN-F's key outputs? There are a number of proofs of DNN's nonparametric consistency for generic architectures -- for instance farrell2021deep. HNN and HNN-F are restricted DNNs, or, alternatively, semiparametric models. If restrictions are approximately true (like the separability in HNN, and the factorization in HNN-F), then we can be confident our $h_{t,j}$'s are close to true latent states. Those restrictions can be implicitly tested by fitting a fully connected DNN with the same data and comparing predictive performance out-of-sample or out-of-bag. Thus, if HNN increases bias much less than it curbs variance, it will supplant the plain DNN. It is interesting to note that the restrictions' benefits are twofold: they reduce variance and provide interpretability.
The proposed strategy to quantify uncertainty is computationally economical since it directly recycles the ensemble, in the spirit of MRFjae for Macro Random Forest. However, it is not devoid of possible deficiencies. First, in an ideal world with more data, the block size should be higher than 2 years to accommodate for highly persistent predictors in HNN's information set. Second, one needs to think carefully about how initialization randomness ends up artificially increasing the reported bands. The good news are that while small networks estimation results depend significantly on randomly drawn weight initialization values, that of large networks do not d2020double. Thereby, quantifying uncertainty directly through the ensemble will provide a reasonable and indicative measure of estimation uncertainty only for large networks (which is being used in this paper). Additionally, HNN uses full batches of data (the whole subsampled datasets for each $b$) rather than stochastic gradient descent. Thus, for each $b$, randomness comes from the estimation subsample, with a minimal contribution from initialization, and little to no contribution from optimization itself. Appendix (ref) features a simulation study with a data-generating process (DGP) plausibly resembling that of the empirical work. It is found that the uncertainty quantification strategy delivers suitable estimates for the kind of (large) architecture considered in Section (ref).
I first analyze contributions series obtained from HNN and HNN-F. Then, I focus on HNN-F which provides a breakdown of contributions in terms of gaps and coefficients. Lastly, I investigate which original inputs contribute most to HNN-F's $g_t$, and how they do so.
As starting point, hemisphere outputs are displayed in Figure (ref) for a training sample ending in 2019Q4. Figure (ref) (Appendix) reports largely unchanged estimates from using a training sample ending in 2007. Lastly, Figure (ref) reports inflation shares in two ways.
First, we observe large positive contributions of $h_{t,g}$ in the 1970s and 1980s which have been much more muted since then, in line with the declining PC narrative (this will be formally assessed when looking at $\gamma_t$ in Figure (ref)). But that was before the pandemic. HNN-F and HNN (Figure (ref)) both report an extremely high positive contribution from $g_t$ to $\pi_{t+1}$ starting from late 2020--as projected from a fixed structure estimated up until 2019. As a result, HNN-F's (and HNN as well) are forecasting annualized headline inflation consistently above 4.5% starting from 2020Q4 (see Figures (ref) and (ref)). HNN's estimation of $h_{t,g}$ seems to align with the growing evidence that the PC is nonlinear in traditional economic indicators space and that the steep part of it has simply been unsolicited in recent decades linde2019resolving,MRFjae,forbes2021low. This will be more formally assessed in Section (ref). Results also cast some doubts on methodologies forcing smoothness through laws of motion. Those typically require potential output to trend upward slowly (a random walk, or local-level process) whereas $h_{t,g}$ has been subject to many upward/downward movements that are much faster than these methods typically allow for.
HNN's successes and failures in forecasting post-2019 inflation can be further understood from Figure (ref). We see that the "overkill" downswing that will later materialize in 2020 inflation forecasts is entirely due to the real activity component. This is because HNN — and Phillips curve regressions in general — are not supplied with external information about the exceptional nature of this recession (a forced partial shutdown of the economy). This makes inference and extrapolation from previous recessions difficult and empirical results for 2020Q1-2020Q3 reflect that. When it comes to the increase in the first half of 2021 is due to a pattern very similar to the 1970s being replicated, that is, a gentle positive impulse from $g_t$ followed by upward pressure from $\mathcal{E}_t^{\text{SR}}$ which eventually settles whereas the forcing from $g_t$ remains.
Contribution of the $\mathcal{E}_t^{\text{SR}}$ component was extremely strong during the 1970s and has been dormant since the beginning of Paul Volker's chairmanship up until early 2021. In 2021, the hibernating $h_{\mathcal{E}^{\text{SR}},t}$ wakes up, and captures the early consequences of supply chain disruptions and the expected consequences of it. It appears that the main reason why inflation forecasts did not climb to 1970s levels in late 2021 is that the overall $h_{\mathcal{E}^{\text{SR}},t}$, despite its earlier spike, shows much less persistence than 4-5 decades ago. Said differently, expectations are still relatively well-anchored, by not deviating persistently from the long-run ones. Additionally, $h_{\mathcal{E}^{\text{LR}},t}$ helps by being almost 2 %-points lower than in the 1970s. Indeed, $h_{\mathcal{E}^{\text{LR}},t}$ is found to be slowly decreasing from the 1980s onward, as expected.\footnote{Note that the overall level of $h_{\mathcal{E}^{\text{LR}},t}$ is not identified separately from $h_{\mathcal{E}^{\text{SR}},t}$ and here it was set by normalizing the other three components to have mean zero over the sample. } Finally, the commodity group (with oil being naturally its most influential member) contributed strongly from the first oil crisis of the 1970s, through the second oil shock, and ends after the second of the twin recessions.
In Figure (ref) shows the decline of the overall influence of $\mathcal{E}_t^{\text{SR}}$ in favor of $\mathcal{E}_t^{\text{LR}}$, with the emergence of trend inflation dominance in the mid-1990s. $\mathcal{E}_t^{\text{SR}}$'s contribution peaks during the 3 inflation spirals of the 1970s. The share of $h_{t,g}$ appears more stable than what is typically reported by PC regressions although it seems to be milder between 2000 and 2020. Lasting effects of energy and commodity prices appear to be gently declining. Figure (ref) makes clear that key historical increases are always due in large part to $\mathcal{E}_t^{\text{SR}}$ up until 2020, when $h_{t,g}$ takes the center stage.
{ \sc Comparison with Traditional Phillips Curves.} Since gaps themselves rather than contributions are what is typically reported, Figure (ref) presents contributions from canonical PC regressions for comparison purposes. In the case of "CBO", those are constructed from a traditional PC specification (including 2 lags of $\pi_t$ and the CBO output gap) with time-varying coefficients obtained from GC2019 two-steps ridge regression approach. Contributions are interesting in their own right because, unlike gaps and coefficients, they are completely identified and expressed in "inflation units". The difference between HNN-F and alternatives is striking for $h_{t,g}$, with the latter giving real activity much less weight in driving inflation than what the former reports. This is especially true in the 1970s and 1980s, but also for recent years. From an ocular spectral analysis standpoint, it is clear that ${h}_{t,g}^{\text{HNN}}$ includes much higher frequencies than raditional gaps/contributions. ${h}_{t,g}^{\text{HNN}}$ is prone to rapid spikes that the alternatives completely forego (e.g., the mid-1980s, the years preceding the 1990s recession,and the mid 1990s).
"HNN-F CBO", which replaces all the activity data in $\mathcal{H}_g$ by the CBO gap itself -- thus keeping all the other modeling ingredients from HNN -- partly helps in understanding this wedge. Indeed, the green and red line follow each other closely except following the 1981-1982 and 2008-2009 recessions. "HNN-F CBO" seems to use nonlinearities to avoid the two very negative contributions from the PC regression. Nonetheless, it is clear that a key difference between HNN and the canonical PC regression is the nonlinear processing of a rich real activity data set. Finally, only the classic PC regression and chan2016's gap (CKP) suggests a deep and lasting negative output gap following 2008. "HNN-F CBO" circumvent this by interacting with a small implicit $\gamma_t$ (to be made explicit in Section (ref)). HNN -F follows a very different pattern where the gap closes rapidly (as early as 2011) but remains gently in negative territory at least until 2018. Finally, from the early 1990s up until the Great Recession (GR), both "CBO" and "CKP" contributions are practically 0 whereas HNN-F sees a mild downward contribution from real activity in the early 2000s.
HNN's post-2020 estimates differ even more from that of standard techniques. CKP's gap in Figure (ref) behaves like most unemployment filtering methods do. It reports strong overheating in the late 2010s\footnote{This is because the trend has been adjusted downward by then. Estimates including COVID-19 observations make this even more pronounced.} and a gently positive gap contribution from late 2021 onward. As we will see in the forecasting results of Section (ref), this will be largely insufficient as an upward forcing to obtain well-centered forecasts during 2021. This is no surprise: this approach yields an output gap which is negative through 2021 and the PC coefficient is small. While persistent gaps are the norm, even in more modern incarnations like hasenzagl2018model, they appear to be difficult to map into the inflation observational record, and provide the well-known assessment of very low passtrough from real activity to inflation.
In Figure (ref), the results for HNN-F and its altered version suggest that the information contained in $\mathcal{H}_\mathcal{E}$ beyond lags of $\pi_t$ only seldom makes a difference --- except for the latest inflation upswing. It is also obvious that baseline HNNs allocate a smaller fraction of inflation to expectations, which is particularly visible from the 1970s inflation spirals (mostly the second) and the 1980s. One way to explain this is that a suboptimal \( g_t \) places an excessive burden of explanation on the lagged values of \( \pi_t \).
{ \sc Robustness to Alternative Separation and Timing Assumptions.} Appendix (ref) evaluates various deviations in the specification of the HNN model to assess if the key finding—that real economic activity has a stronger effect on inflation than captured by standard Phillips curves—still holds. First, by specifying short-run expectations in a tighter way (using only official expectations data), the study found that the more cleanly separated HNN yields very similar results on the importance of $g_t$. Second, the timing of target variables is changed to align with the theoretical NKPC (a contemporaneous relationship) and design in studies such as blanchard2015inflation and coibion2015phillips. Although $h_{t,g}$ show some quantitative differences in the post-2020 sample, the historical path, the resulting post-pandemic narrative, and its overall significance remains consistent with baseline results. Third, a different specification of long-run expectations ($\mathcal{E}_t^{\text{LR}}$) also maintains the core findings. This suggests that while compositional assumptions and data adjustments can influence the interpretation of short-run and/or long-run expectations, they do not significantly impact the estimated role of real activity in driving inflation.
So far, the focus has been on $h_{t,g}$. As discussed in Section (ref), HNN-F, at the cost of additional assumptions, allows for a separate inspection of $g_t$ and $\gamma_t$. Figure (ref) reports them for estimation ending in 2019Q4.\footnote{Figure (ref) reports the other two coefficients, which have heterogeneous shapes, implying HNN-F is not merely "smoothing" through inflation. } Unlike recessions that preceded it, the GR is characterized by a rapid yet incomplete closing of the gap. Interestingly, this mildly negative gap lasting for a decade coincides in part with the so-called missing inflation era. This observation -- a rapidly closing gap followed by a long slightly negative one -- is found whether I estimate the model using data up to today, or end estimation in 2007. Thus, there is no indication for a reverse engineering of $g_t$ to fit the post-GR inflation data. Moreover, the rapid closing of \(g_t\) following the Great Recession is not observed for the early 1990s and 2000s recessions. This observation suggests that the fast closing of \(g_t\) after the GR is not a mechanical feature of HNN.
Let us now consider \(\gamma_t\), the widely studied evolving coefficient of the Phillips curve. The evidence in Figure (ref) is in partial agreement with the recent literature on the matter blanchard2016phillips,gali2015monetary,del2020s in the sense that the exogenously time-varying $\gamma_t$ has been decreasing starting from the 1980s. However, there are many notable differences. First, there seem to be a break around 1980, in the midst of Volker's disinflation, where $\gamma_t$'s decline substantially accelerates. Second, unlike results from standard approaches, $\gamma_t$ is not found to decline further following 2008, but rather to slowly come back to its former value starting from the 2000s. These observations differ from those in studies such as blanchard2016phillips, stock2019slack, and hazell2020slope. Despite their diverse research designs, these studies commonly use forcing variables that are closely correlated with filtered unemployment. Given how different HNN-F's $g_t$ is with respect to traditional indicators, $\gamma_t^{\text{HNN}}$ atypical vivacity is not entirely surprising -- and turns out to be in line with other recent works. bonam2021time report that using a survey-based labor shortage indicator is a better predictor of wage inflation in many European countries and leads to stable or steepening PCs over the last decade. Recently, domash2022tight report results in agreement with this view using US state-level data.
For space considerations, many results have been relegated to the appendix. In Appendix (ref), I report $g_t$ for various estimation windows and find that it is desirably stable across those. In Appendix (ref), I report gaps and coefficients using two alternative supervisor variables (Core CPI and Yearly Inflation).
By construction, $g_t$ and $\mathcal{E}_t^{\text{SR}}$ are combinations of thousands of parameters nonlinearly processing many regressors. Consequently, directly looking at network weights is inevitably meaningless. More productively, I investigate which $X_{t,k}\in \mathcal{H}_g$ seems to matter most by designing a variable importance (VI) exercise very much inspired from what MRFjae studied for "generalized time-varying parameters" in a Random Forest context -- which is itself inspired from traditional variable importance measures for tree ensembles predictions. For the VI calculations, I focus on groups of variable $k$, meaning we will evaluate the overall effect of all transformations and lags of variable $k$. Implementation details can be found in Appendix (ref).
VI results are reported in Figure (ref). Here are key observations for $\text{VI}^g$. First, AWHMAN's (average weekly hours in the manufacturing sector) predominance for $g_t$ suggests an important role for the intensive margin, whereas typical labor-based gap measures are mostly about extensive margin (like filtered unemployment). Recently, bulligan2019adjustments find the former can complement the latter as forcing variables in linear PCs for the Euro area (but not the US). Second, the composite Help-Wanted Index (HWIx) of barnichon2010building --which mccracken2020fred splice earlier in the sample with the original Conference Board product for "print" job postings -- is shown to play an important role. Intuitively, the index, by construction, characterizes increased labor demand (and perhaps shortages) which is expected, by economic theory, to translate into higher wages, and eventually, higher aggregate prices. This partly explains the very positive gap in Figure (ref) since HWIx is effectively skyrocketing as of late 2021 despite being largely stagnant in 2018 and 2019.\footnote{ domash2022tight also find that indicators associated with the demand side of the labor market have better predictive power for wage inflation. Similarly, bonam2021time document that a survey-based labor shortage indicator is a better predictor of wage inflation in many European countries than an unemployment gap.} However, this not the whole story: nonlinear neural processing of HWIx seems essential as reported in the ablation study (Appendix (ref)). In Figure (ref) (Appendix), we see that, at times, the unemployment rate and HWIx were closely related, like during 1990s and the 2000s. But other times they were not, like the 1970s and during recent years. Moreover, their acceleration rate can differ in key recession and expansion episodes. By betting on some transformation of \texttt{HWIx}, HNN leveraged historical patterns to rely on more potent (and timely) forcing variables. Third, GDP and associated measures seem unimportant, so does the unemployment rate. The only traditional gap measure appearing in the lower end of $\text{VI}^g$'s top 25 is total capacity utilization (\texttt{TCU}).
Turning to $\text{VI}^{\mathcal{E}_\text{SR}}$, we see that a handful of very familiar variables dominate the top 25. First, the obvious preponderance of the University of Michigan Survey of Consumer Inflation Expectations (inf_mich) strengthens the case for the increasingly popular practice of using survey expectations in PC regressions binder2015whose,coibion2015phillips,coibion2018formation,meeks2023heterogeneous. It also completes the explanation as to why HNN-F forecasts did not call for lasting disinflation following the GR. That is, as suggested by coibion2015phillips, proxying expectations using survey expectations rather than, say, lags of the CPI, procures more accurate post-2008 predictions. HNN learned that prior to 2007 by putting a high weight on inf_mich. Nonetheless, $\text{VI}^{\mathcal{E}_\text{SR}}$ suggests mixing in expectations from different economic agents and formulated for different horizons seems more appropriate, which is in line with recent results for simpler regression models in banbura2021inflation. There is also a minor role for "backward-looking expectations" or "inflation persistence" as emphasized by the presence of lags of the CPI (Ylag) in the top 4.
The black box can be opened further. Now that we identified important inputs for $g_t$ and $\mathcal{E}_t^{\text{SR}}$, we can use a surrogate model approach as in MRFjae to map back the latent series of interest into pre-identified key inputs. The surrogate model approach, which consists of fitting an auxiliary interpretable model to a pre-estimated function output often using a focused set of inputs molnar2019interpretable, is an economical way to asses the shape of nonlinearities. The surrogate model used below is a bivariate Random Forest featuring, for both $g_t$ and $\mathcal{E}_t^{\text{SR}}$, the two most salient inputs identified by VI. While RF is typically considered a black box model, its fit is easy to visualize if the inputs are 2-dimensional. Moreover, unlike single trees (often used as surrogate devices), it can approximate hard-thresholding and smooth relationships. Implementation details can be found in Appendix (ref).
The VI analysis above suggests that key drivers of $g_t$ are vacancies and hours worked, and those of $\mathcal{E}_t^{\text{SR}}$ are professional forecasters' (spf_cpi_currentYrs) and consumers expectations (inf_mich).\footnote{While Ylag comes in before spf_cpi_currentYrs, the combined importance of the two SPF indicators is greater than the lags of CPI. Results are mostly unchanged from using spf_cpih1 instead of spf_cpi_currentYrs. } Figure (ref) reports results on how the two selected indicators are associated with $g_t$ and $\mathcal{E}_t^{\text{SR}}$. The correlations of the surrogate models' fitted values with the original series are 0.80 for $g_t$ and its reconstruction, and 0.77 for $\mathcal{E}_t^{\text{SR}}$ and its reconstruction, suggesting that they capture a significant part of the relevant variation in $g_t$ and $\mathcal{E}_t$.
The real activity relationships show significant nonlinearity, particularly with respect to vacancies. There is a notable kink in the behavior of vacancies: the effect is moderately positive below a certain threshold, but beyond this threshold, the derivative increases substantially, causing the curve to become very steep until another plateau is reached. The effect of hours worked is linearly positive when vacancies are low but becomes quite nonlinear at higher vacancy levels, showing a major increase from low to mid-range values. When both hours worked and vacancies are high, the effect is milder than for mid-range values. However, it is important to note that there are very few data points in the region where both vacancies and hours worked are at the top of their range, so the second segment of the observed U-shape should be interpreted with caution. This kink behavior, or hockey stick relationship, between vacancies and $g_t$ is reminiscent of the nonlinear Phillips curve proposed by benigno2023s, who propose an economic rationale for the phenomenon. While less transparent than a tightly specified theoretical model, HNN identifies this behavior among many possible nonlinearities from data through 2019Q4.
Overall, the relationships between expectations series and $\mathcal{E}_t^\text{SR}$ show some nonlinearities. However, these can be reasonably well approximated with non-interactive linear relationships. This suggests that while there are deviations from linearity, especially for lower values (see ablation study in Appendix (ref)), the link between survey-based expectations and inflation appears less complex in terms of functional form than that of real activity.
The highly nonlinear relationship behind $g_t$ and the semi-linear ones behind $\mathcal{E}_t^\text{SR}$ provide further backing for this paper's core finding about the importance of real activity. Much of the prior evidence is based on linear models, which, by construction, attribute larger explanatory power to relationships they can capture well—in this case, expectations rather than real activity. In a more general nonlinear model, $\mathcal{E}_t^\text{SR}$ remains very important. However, $g_t$ becomes equally significant, and even more so for the post-2020 sample, once the model accounts for the nonlinearity in how a diverse basket of real activity indicators can drive inflation.
In Appendix (ref), I report results for a linear version of HNN where ReLU activation functions have been replaced with linear ones. The linear model struggles to capture a plausible path for $g_t$, whereas results for short-run expectations are closer to those obtained by the baseline HNN. This reinforces the view that allowing for nonlinearities is of greater importance when modeling real activity. Additionally, the forecasting performance of the linear version (included as a benchmark in Section (ref)) is substantially inferior to that of HNN and HNN-F, especially in recent years.
Current evidence in favor of PC-based forecasting is weak, with minor or non-existent improvements over simpler benchmarks like plain autoregressions atkeson2001phillips,stock2008phillips,faust2013forecasting. Recent extensive evaluations for the Euro area banbura2020does suggest there is a case for some cautious hope with specifications allowing for flexible trend inflation and an endogenously estimated gap. Despite all the evidence on its uneven empirical potency, PCs are still widely used to forecast and understand inflation yellen2017inflation, mostly because they are rooted in some basic form of macroeconomic theory. This section provides a validation of the new methodology by comparing its predictive performance to a representative set of traditional econometric approaches and machine learning tools.
{ \sc Setup.} The pseudo-out-of-sample period starts in 2008Q1 and ends in 2024Q1. I use expanding window estimation from 1961Q3. HNNs are re-estimated and tuned every 4 quarters. Following standard practice, the quality of point forecasts is evaluated using the Root Mean Square Error (RMSE). The forecasting target is CPI($s=1$), which is the supervisor in benchmark HNN specifications. Additionally, results for two alternative supervisors are studied in Appendix (ref) -- CPI average inflation from $t$ to $t+4$ (${\pi}_{t:(t+4)}=\sum_{s'=1}^4 \pi_{t+s'}$) and Core CPI($s=1$). Performance results are reported for four subsamples. First, we have all the samples including and excluding 2020 observations.\footnote{The exclusion zone is extended to 2021Q1-Q2 for 4 quarters ahead forecasts for the simple reason that they were made during the depth of 2020Q1-Q2 and the models propagate a year later what it thinks is an unusually large negative (yet typical in composition) demand shock.} Additionally, separate RMSEs for the pre-2020 (up to 2019Q4) and post-2020 (starting in 2021Q1) sample are provided.
{ \sc Models.} A few obvious benchmarks from both sides of the aisle are considered. On the ML side, there is a fully connected neural network with the same hyperparameters as HNN (DNN) and a Random Forest (RF) with default tuning parameters (typically hard to beat, MRFjae). Additionally, I consider a linear network version of HNN-F (HNN-F-LR) where ReLU activation functions have been replaced by linear ones for gaps hemispheres. Using linear layers in the network makes it some kind of linear groupwise supervised principal components model, a linear data-rich benchmark. The latter model allows to get a sense of the contribution of nonlinearities in HNN's overall performance by ablating them. All these benchmarks use the exact information set as HNN (variables and aforementioned transformations).
Then, there are inflation-specialized econometric benchmarks of increasing sophistication. First, we have the AR(4) which will stand as the generic numeraire of reported MSEs. Also, two rolling means are considered, the one-year mean à la atkeson2001phillips (1y Avg) and a longer-run one (10y Avg). Bringing in real activity information, I consider a PC regression (PC, two lags of $\pi_t$ and the CBO gap) estimated on a rolling window of 15 years to allow for time-varying parameters. Note that this PC regression is given a handicap by using the latest CBO gap which may have been substantially revised ex-post. Additionally, an identical PC regression augmented with two lags of oil prices and survey expectations (PC+) is considered to match some of the information set in HNN, and more generally specifications inspired from coibion2015phillips. We also consider chan2016's time-varying bounded Phillips curve model (CKP) where $g_t$ is extracted in a supervised fashion from unemployment by assuming the natural rate of unemployment to follow a random walk. Key coefficients also follow random walks. This approach was reported to have sporadic success in forecasting Euro area inflation banbura2020does. All those non-NN methods are re-estimated every quarter.
{ \sc RMSEs and Forecasts Narratives Comparison.} I now report the forecasting performance of HNNs for the three targets and examine their forecasts. In Figure (ref), HNN and HNN-F perform well for CPI ($s=1$), excluding the aberrant 2020 observations (to which we will come back to below). Figure (ref) shows that HNN's relative success is partly due to its reasonable accuracy in capturing the recent upswing in inflation. Overall, the bar plot in Figure (ref) show improvements ranging from 10% to 25% depending on the subsample, with HNN-F and HNN consistently delivering comparable RMSEs.\footnote{Diebold-Mariano test statistics are reported in Table (ref) (Appendix (ref)) and show that in the main case of CPI ($s=1$), these improvements over the benchmark are significant at the 5% level for the "Without 2020" and "Post 2020" samples. Table (ref) and Figure (ref) show that forecasts using real-time data (and resulting raw RMSEs) are extremely similar to the main ones in Figure (ref) using the 2024Q1 data.} The closest competitors are RF, PC+ (which includes the survey of professional forecasters' forecasts as a predictor), and CKP. The mild improvement of HNNs over (or tie with) benchmarks for the pre-2020 sample can be understood from, e.g., Figure (ref), as all models agree on a near-zero contribution of real activity from 2010 to 2020. However, traditional models are less apt at capturing what crucially matters for policy: when inflation starts to persistently get out of its target range. For the three years following 2020 (Post 2020 in green), HNN, and particularly HNN-F, perform best for CPI ($s=1$) and the two additional targets in Appendix (ref). Finally, the linear supervised data-rich model (HNN-F-LR) struggles in most samples and highlights the importance of nonlinearities to truly harness the benefits of the additional data—an argument made more generally for ML models in macroeconomic forecasting by GCLSS2018,GCMS.
Figure (ref) helps differentiate the narratives. CKP, based on a Bayesian bivariate state-space model of trend inflation and the gap, faces major difficulties during two important historical episodes. First, its forecasts are consistently too low for most of 2008-2012, exemplifying the missing disinflation puzzle—a classic issue with regression models predicting inflation using traditional gaps. HNN mostly avoids this predicament with a rapidly closing gap in the aftermath of the GR. Second, CKP forecasts are significantly below realizations for the two-year period from mid-2020 to mid-2022 (when inflation starts to slow down). The reason for this is visible in Figure (ref): the de-trended unemployment rate, the forcing variable, is negative for most of 2021. Consequently, if it influences inflation, it does so downward, not upward. When the unemployment-based forcing variable finally lands in positive territory, the Phillips curve coefficient makes its contribution minimal. PC+ also struggles, but less so than CKP, to swiftly and consistently capture the sudden rise in inflation in 2021, which settled around 5% rather than 2-3%. HNN delivers more timely forecasts of CPI ($s=1$) in the post-2020 sample by not relying on potentially lagging indicators and leveraging nonlinearities to capture the contribution of real activity in a timely manner.
{ \sc On 2020's Performance and Density Forecasting.} Unsurprisingly, yearly results for 2020 and most of 2021 are not great for real-activity-based forecasts, including HNN. In a similar fashion to what is reported in Figure (ref), this is due to HNN and PC regressions not being informed that this is no ordinary recession and that extraordinary governmental programs have been implemented to life support the economy. This limitation has even stronger consequences when forecasting ${\pi}_{t:(t+4)}$ since the medium-run dynamic transmission mechanism itself is certainly quite different during the pandemic than for previous recessions. A careful use of the model, combined with the usual external judgment in forecasting practice, would have led to discarding the downward spike. Therefore, it is justified to focus on performance metrics excluding them. Nonetheless, this setback motivated the development of an extended version of HNN in GCFK which features an additional hemisphere predicting forecast uncertainty. The obtained density forecasts are very competitive (including 2020) because the more sophisticated network calls for extreme volatility during 2020 (thus, self-discounting its struggling 2020 forecasts) and more moderate volatility afterwards.
This paper estimates a neural Phillips curve with a deep output gap using the Hemisphere Neural Network (HNN), a machine learning model designed for economically interpretable inflation predictions. HNN avoids various non-innocuous assumptions of traditional econometric methods, leading to significantly different results. A refinement of the plain HNN architecture allows for the separation of contributions from expectations and real activity into coefficients and gaps.
HNNs provide good forecasts, such as capturing part of the CPI upswing in 2021. This success is due to HNN defining an output gap through supervised nonlinear processing of numerous real activity indicators. The most important raw indicators are vacancies and hours worked, suggesting a prominent role for the intensive margin and the demand side of the labor market. Allowing for nonlinearities is crucial in capturing such effects. As a result, the Phillips curve coefficient on HNN's output gap declined in the early 1980s but experienced a significant revival starting from the 2000s. The role of real activity, found to be significantly stronger than in traditional Phillips curves, explains HNN's ability to foresee mounting inflationary pressures from late 2020 onward.
The HNN framework is versatile and can be applied to various problems and extended in many directions. Overall, this paper demonstrates that deep learning can provide more than just predictions, helping to address key issues in empirical macroeconomic analysis, such as the successful measurement of ambiguously defined yet extremely important latent states.
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