1. State-Space Forms – Semi-structural and DSGE Models

Formulating potential output estimators in a state-space form as linear filters is surprisingly simple. This is the case since the celebrated Kalman filter, see e.g. Kalman (1960) or Whittle (1983), originates from the Wiener filtering theory and deals with an important special class of stochastic processes.

The state-space formulation of the potential output estimation became very popular, partly due to its flexibility, see e.g. Kuttner (1994), Laubach and Williams (2003), inter alios. The state-space model is easy to formulate and modify, easily handles missing data or non-stationary dynamics, and is a natural representation for linearized recursive dynamic economic models.

The missing piece in the literature on the multivariate model analysis is the explicit acknowledgement and use of the fact that the Kalman filter and smoother³ are actually just that – filters. As demonstrated above, an explicit linear filter formulation is useful for obtaining a decomposition into observables. The formulation of the state-space model as a linear filter, the filter weights, and a very practical implementation of decomposition into observables for state-space models, follow.

Filter representation For the purpose of analysis, it is assumed that the model takes the following state-space form:

Here ε ~ N(0, Σ_ε), Σ_ε = I and thus structural shocks are not correlated with measurement error shocks, with no loss of generality. The vector of transition, or state, variables is denoted by X_t, whereas observed variables are denoted by Y_t. By imposing a restriction that GΣ_εH′ = 0, it is guaranteed that measurement errors and structural shocks are uncorrelated.

The state-space model (9)–(10) can be used to estimate the unobserved states and shocks {X_t, ε_t} from the available observables {Y_t}. The output gap is one of the elements in X_t. I shall focus mainly on the ‘smoothing’ case, i.e. when estimates of X_t are based on observations available for t = 0,…T, using the notation 𝔼[X_t|Y_T,…, Y₀] = X_t|T.

In the case of multivariate models with multiple observables, the possibilities for analysis are richer than in the case of univariate models. If meaningful, the exploration of impulse-response and transfer functions provides insights regarding the model properties, together with the popular structural shock decomposition. In other words, if the shocks have some structural interpretations, one can express the observed data (and unobserved states) as cumulative effects of past structural shocks,

where A(L) = (I − TL)⁻¹G and ${\hat{\epsilon }}_t,{\mathbf{X}}_{t|\infty }$ denote a mean squared (Kalman smoother) estimate of the structural shocks, ε, and state variables X.⁴ Expression (11) is frequently used in a DSGE analysis for storytelling and interpretation of macroeconomic data.

Now is the time to reverse the logic and ask the question, “What observed data drive each particular unobserved structural shock and state variables?” That is the purpose of this paper – to draw a closer attention to a presentation of the unobserved state estimates as a function of the observed data. In the case of a doubly-infinite sample the model can be expressed as

In real world applications, where the sample is always finite, the optimal finite sample implementation of (12) leads –or at least should lead– to a multivariate linear filter with time varying weights,

Here the weight sequence varies with every time period t. That is because the Kalman smoother carries out an optimal mean-square approximation of the infinite filter $\mathbf{\Omega }\left(z\right)={\mathrm{\Sigma }}_{t=-\infty }^{\infty }{\mathbf{\Omega }}_j{z}^j$ with a finite length filter $\mathbf{\Omega }\left(z\right)={\mathrm{\Sigma }}_{i=t_0}^T{\mathbf{\Omega }}_{j,t}{z}^j$, so as to minimize the distance ||X_t|∞ − X_t|T||². It operates under the assumption that the model (9)–(10) is the data generating process for the data. More on this in a discussion of real-time properties of output gap estimates below.

To decompose the output gap into observables and to analyze data revisions using (13), it remains to either calculate the time varying weights of the filter or to reformulate the problem such that one can avoid the computation of weights. Luckily, both options are readily available and their description follows.

Weights of the Filter Weights of the filter Ω(L) are not data-dependent and are a function of the model specification only.⁵ In the case of the doubly-infinite sample, the Wiener-Kolmogorov formula, see Whittle (1983), implies that Ω(z) = Γ_XY(z)Γ_Y(z)⁻¹. Here Γ_Y(z) and Γ_XY(z) stand for auto-covariance and cross-covariance generating functions of the model:

The transfer function of the model, Ω(z), is the key ingredient for a frequency-domain analysis of the filter. In the Applications section below, I explore transfer function gains for a semi-structural model of the output gap, which indicates the most relevant frequencies of observed time series for the output gap estimate. The core of the analysis is in time domain and thus the weights are needed.

For all but very simple and small models, it is difficult to get an analytical description of the weights in (12) using the transfer function of the model. The weights can, however, always be obtained numerically. An inefficient, but operational way would also be to compute the inverse Fourier transform of (12). Koopman and Harvey (2003) provide a recursive way to calculate time-varying weights in (13) for general state-space models and a lucid paper by Gomez (2006) provides time-domain formulas to calculate weights in (12).

In particular, Gomez (2006) shows that for the model (9)–(10) the weights Ωj, adjusted to model notation above, follow as

where L ≡ T − KZ, K denotes the steady-state Kalman gain and P is the steady-state solution for the state error covariance given by a standard discrete time algebraic Ricatti equation (DARE) associated with a steady-state solution of the Kalman filter. R_|∞ is a solution to the Lyapounov equation, R_|∞ = L′R_|∞L + Z(ZPZ′ + HΣ_εH′)⁻¹Z′, associated with the steady-state Kalman smoother solution. R_|∞ is the steady-state variance of the process, r_t|∞, in the backward recursion, X_t|∞ = X_t|t−1 + Pr_t|∞, where in finite-data smoothing r_t−1 is a weighted sum of those innovations (prediction errors) coming after period t − 1. Finally, Σ = Z(ZPZ′ + HΣ_εH′).⁶

The relationship between the time-invariant weights of the filter and the time-varying weights, used in the case of the finite sample implementation by the Kalman smoother, is unique and is discussed below in the section devoted to real-time properties and news effects. The intuition behind the re-weighting is simple, though. The Kalman smoother implementation implicitly provides optimal linear forecasts and backcasts of the sample and applies the convergent time-invariant weights.

Practical Implementation of the Decomposition into Observables To compute the observable decomposition one can always calculate the weights using Koopman and Harvey (2003) recursions and implement the moving average calculations. That requires calculating and storing large objects, pre-programmed tools, or a little bit of advanced knowledge of state-space modeling. Sometimes, time constraints might prevent analysts from using these tools. Having a shortcut is thus beneficial.

A particularly simple and accurate way is to view the Kalman smoother as a linear function of multiple inputs, denoted by X = F(Y), where X and Y are (T × n) and (T × m) matrices. The great thing about that function is that it is linear. For stationary processes, the Kalman smoother provides the least squares estimates of the form $\mathbf{X}=\hat{\mathbf{\Omega }}\mathbf{Y}$, where $\hat{\mathbf{\Omega }}$ is the matrix of time-varying filter coefficients. Trivially, for two different sets of observables, {Y^A, Y^B}, one obtains ${\mathbf{X}}^{\mathbf{A}}-{\mathbf{X}}^{\mathbf{B}}=\hat{\mathbf{\Omega }}\left({\mathbf{Y}}^A-{\mathbf{Y}}^B\right)$. By appropriate non-overlapping grouping of differences in inputs, one can easily obtain the effects of the change of measurements on all estimated unobservables and carry out the decomposition analysis. This method works for any model with two different sets of observables and a common initial state, unless the change in the initial state is treated as well. There is no need to know the values in $\hat{\mathbf{\Omega }}$; the whole decomposition of the deviations in the two estimates can be obtained by successive runs of the Kalman smoother with different inputs.

The two data input structures can bear many forms. The only requirement is that the structure of observations in both datasets must be identical. This setup is very feasible in the case of data revision analysis and exploration of the effects of new observations, as shown below. The counter-factual dataset can also take form of a steady-state (balanced growth path), unconditional forecasts, etc. – depending on the goals of the analysis. Decomposition into observables in this paper is consistent with a dataset featuring missing observations and direct observations implementation of linear restrictions, often used for imposing the ‘expert judgement’.

2. Univariate filters – Band-Pass, Hodrick-Prescott, etc.

True, decomposition into observables in a univariate setting is not an issue, as the only variable that enters the output gap estimation is the output itself. The analysis of their real-time properties and news effects, however, follows the same principles as in the case of multivariate filters. A proper understanding of frequently used univariate filters, such as band-pass filters or the Hodrick-Prescott filter, is important, as these often form parts of multivariate models.

Univariate filters are specified either directly in terms of their weights in time domain, directly in terms of their transfer function in frequency domain, as a state-space model, or as a penalized least squares problem – e.g. the Hodrick-Prescott filter or exponential smoothing filter. Being specified in any of these ways, they have a time domain filter representation as

where w_i are the weights of the filter. This fact is well know and is restated just for clarity and completeness. Univariate filters are usually discussed in terms of their spectral properties, implied by F(z), but the weights of the filter are sometimes discussed as well, see Harvey and Trimbur (2008), among others.

Most contributions to the literature focused on designing or testing the univariate filters. They are concerned with (i) approximation of the ideal band-pass filter or (ii) revision properties of the filters for increasing sample size. The ideal band-pass filter with a perfectly rectangular gain function is often considered as a natural benchmark to judge univariate statistical filters against in terms of their ‘sharpness’ – i.e. leakage, strength of the Gibbs effect, or ease of finite-sample implementation.

The class of linear filters is large, including a variety of bandpass and highpass filters. Apart from the Hodrick-Prescot/Leser filter, Butterworth filters analyzed in Gomez (2001), rational square wave filters suggested by Pollock (2000), or even multiresolution Haar scaling filters fit the representation (19).

Given a possibly infinite filter F(z), the revision properties depend on the quality of it’s finite sample approximation, which is crucially dependent on the data generating process of the data, see Christiano and Fitzgerald (2003) or Schleicher (2003). Discussion of the revision properties and optimal finite sample implementation is discussed below, since univariate filters are often part of semi-structural multivariate models or production function approach estimates of the output gap.

3. Structural VARs

Structural VARs have a natural moving average, linear filter representation. It seems therefore very desirable to express the potential output and the output gap as a linear combination of data inputs. A thorough analysis of the contributions of observed series and the SVAR estimates frequency transfer function is crucial as these often are outliers in comparison with other methods, see McNellis and Bagsic (2007), Cayen and van Norden (2005) or Scott (2000), among others. Although SVAR models may seem to be used less frequently for output gap estimation, they certainly belong in the toolbox of many central banks and applied economists.

Assume that an estimated reduced form VAR model of order p is available, that is,

where residuals, ε_t (reduced-form shocks) are linked to ‘structural’ shocks η_t via an invertible transformation ε_t = Qη_t. I assume that the dimension of Y_t is n. The identification often imposes long-run restrictions following Blanchard and Quah (1989) to tell apart transitory and permanent component of output, see e.g. Claus (1999). The structural VAR model is then expressed as Y_t = B(L)QQ⁻¹ε_t = S(L)η_t.

Assume that the j-th component of the data vector, Y_t, is the GDP growth, Yj,t = Δy_t, then

and the output gap, x_t, is the part of the GDP not affected by permanent shocks

assuming ${\mathrm{\Sigma }}_{k=0}^{\infty }{\mathbf{S}}_{12}\left(k\right){\eta }_{2,t-k}+\dots +{\mathrm{\Sigma }}_{k=0}^{\infty }{\mathbf{S}}_{1n}\left(k\right){\eta }_{n,t-k}=0$. The structural shocks are estimated from the reduced form VAR residuals using η_t = Q⁻¹ε_t = Q⁻¹A(L)Y_t. One can thus recover the estimated output gap as a function of observations.

The identification scheme itself can be very case-specific, yet it is clear that for SVAR estimates of the output gap their concurrent estimates and final estimates coincide, unless an extended sample is used for paramter re-estimation. Investigating the spectral properties of S(z) is advisable, since ex-ante it is not clear which frequencies of observed series are used for estimation and SVAR estimates often stand out as outliers. The decomposition of the output gap into observables can be done using the expressions above, where the output gap is a function of structural shocks, which themselves are a function of observed data.

4. Production Function Approach

Even a production function (PF) approach can often be expressed as a filtering scheme. The real-time revision properties then crucially depend on the filter representation, as in the case of other methods. Many practitioners are, perhaps, aware of the production function approach being very much dependent on the underlying filters used in various steps of the method. This section provides an explicit formulation of the production function output gap estimate as a linear filter, along with its structure and decomposition into observables.

Assume that the value added is produced using the Cobb-Douglas production function. Denoting the logarithms of individual variables by lower-case letters, one gets

Here a_t is the ‘Solow residual’ and k_t and l_t denote the actual levels of the capital stock and hours worked in the economy. It is common that a trend total factor productivity, $a_t^{*}$, is identified from the Solow residual using some smoothing procedure, often a variant of the Hodrick-Prescott or an other symmetric moving average filter. Denoting the smoothing filter as A(L), it is clear that

The next step is usually a determination of an ‘equilibrium’ or a trend level of hours-worked, or employment, and of the capital stock. I will consider only the estimate of the equilibrium employment, which is often cast as a filtering problem for the NAIRU, frequently in terms of inflation, capacity utilization, or other variables. Importantly, the sub-problem is most often a linear filter.

With only little loss of generality, I assume that the equilibrium employment is given by the trend component of the employment, obtained using the univariate filter as $l_t^{*}=E\left(L\right)l_t$. The output gap, x_t, can then be expressed then as

which is a version of a multivariate linear filter with three observables, or signals.⁷ It is interesting that in this simple case, if the trend component of the Solow residual and of the employment are obtained using the same procedure, then E(L) = A(L) and the contribution of observed employment data gets eliminated. Further, given the fact that the capital stock usually has little variance at business cycle frequencies–as a slow moving variable– its ‘gaps’ tend to be small. The smaller they become, the more the production function approach to output gap estimation approaches a simple univariate filter estimation of the output gap.⁸

In the case where A(L) and K(L) are transfer functions of the Hodrick-Prescott filter, which is quite usual, the production function approach results tend to be quite similar to HP filter estimates and suffer from most of the problems usually associated with the HP filter approach. See Epstein and Macciarelli (2010) as an example.

When the production function approach estimate can feasibly be expressed as a linear function of its inputs (e.g. output, labor, capital stock, etc.), providing such a decomposition is highly desirable. A finite-sample version of (25) is easy to obtain as long the process is linear. The production function estimates are often decomposed into the contributions of the total factor productivity, equilibrium employment, and capital stock, which are not all directly observable.

Incorporating any non-causal filter into the production function approach calculations impacts its real-time, revision properties. The linear filter analysis for revision properties and news effects thus relates also to the production function approach to potential output estimation.

1. Decomposing Effects of Data Revision

By data revisions, only the revision of past data releases should be understood; most often these concern GDP data. The treatment of data revisions is very simple as long as the filtering problem is applied to the same sample and number of observed data series – revised and old, say Y^A, Y^B. The revision and its decomposition into factors is then given by

Again, it is quite useful to think of (29) in a stacked form. For the Kalman filter, which is a solution to the least square problem, one has that $\mathbf{X}=\hat{\mathbf{\Omega }}\mathbf{Y}$ and thus trivially ${\mathbf{X}}^{\mathbf{A}}-{\mathbf{X}}^{\mathbf{B}}=\hat{\mathbf{\Omega }}\left({\mathbf{Y}}^A-{\mathbf{Y}}^B\right)$. The only requirement is to keep the structure of $\hat{\mathbf{\Omega }}$ fixed, for which an identical structure of observations (cross-section and time) is needed. This stacked, matrix representation of the filter is also useful for investigating news effects, treatment of missing variables and filter tunes (linear constraints) after a simple modification, see below.

The data revision decomposition is useful not just for the output gap estimates but also for understanding the revised estimates of technology, preference, and other structural shocks in a forecasting framework based on a DSGE model, see Andrle and others (2009b). For instance, the interpretation of inflationary pressures in the economy changes when the domestic absorption is revised, in the context of unchanged estimates of the CPI inflation.

2. News Effects and End-Point Bias

Implementation of a doubly-infinite, non-causal filter Ω(z) is problematic, since all economic applications feature finite samples. Newly available data lead to some revision of past estimates of the output gap or other unobserved variables. Even in the case of an optimal finite sample approximation of the infinite filter, increasing the sample size leads to revisions.

All non-causal filters suffer the from finite-sample problem and saying that state-space models do not would not be correct. State-space models using the Kalman smoother are no exception to the rule. The statement by Proietti and Musso (2007) that their state-space signal extraction techniques ‘do not suffer from what is often referred to as end-of-sample bias’ is thus incorrect.

Still, the double-infinite sample size formulation of the problem is the best starting point for the analysis of news effects, restated here for convenience:

The revision due to availability of new observations after period T, chronologically only, is then obviously

which is just a weighted average of prediction errors, conditioned on the information set to period T, given the constant weights filter. See Pierce (1980) for a discussion of revision variance in time series models. Population revision variance can be computed given the filter and the data-generating process for the data, determining the prediction errors.

Intuitively, (i) the smaller the weight on future observations and (ii) the better the predictions of the future values, the smaller the revision variance. The discussion below conditions on a model as given and does not put forth advice on how to design a better filter with different weights.

All output gap estimation methods considered in the paper adopt a solution to the infinite-sample problem, either an explicit or an implicit one. Implicitly, all provide forecasts and backcasts for the actual sample, if needed. A simple truncation of the filter weights can be interpreted as zero-mean forecasts, which would be suitable only for an uncorrelated zero-mean stationary process. The optimal finite sample implementation of the filter is a solution to the following approximation problem:

see Koopmans (1974), Christiano and Fitzgerald (2003), or Schleicher (2003) for details. The solution delivers a sequence of filters ${\hat{\mathbf{\Omega }}}_t\left(z\right)$ that, based on position in the sample, t, are the weights of the time invariant filter Ω_t(z), adjusted by a factor derived from the auto-covariance function of the data.⁹

Importantly, the solution to a problem in (32) is equivalent to the one where the time invariant infinite filter is applied on the available sample padded with forecasts and backcasts by the j-step ahead, with j chosen such that the filter converges. The forecast is uniquely pinned down by the auto-covariance function of the data generating process. Hence, a heuristic solution adopted by practitioners is actually the optimal one.¹⁰

State-space models with the Kalman smoother and Penalized Least-Squares formulations provide implicitly the optimal finite sample approximation to Ω(z) by re-weighting the time-invariant weights. This is equivalent to providing backcasts and forecasts and padding the sample. The crucial point is that the forecasts are based exclusively on the covariance generating function of the underlying model, which may not always represent well the covariance structure of the data generating process. The mismatch between the model and the data results in poor forecasting properties. When the weights of the filter are spread out to many periods, poor forecasting properties translate into revision variance and the so called end-point bias.

It is often the case that the filter puts a large weight on observations at the end of the sample. This is simply a consequence of the fact that for most models their forecasting formula puts larger weight on the most recent observations. When a doubly-infinite filter is applied to a padded sample, the end sample observations are implicitly counted many times due to the chain rule of projections, receiving effectively a larger weight.

Example Consider the Hodrick-Prescott/Leser filter as an example. The filter is often mentioned for its poor revision properties, or its large ‘end-point’ bias. Unless one provides a factorization of the filter representation, like Kaiser and Maravall (1999), the filter takes many periods to converge – more than 20 data points on both sides, see Fig. 1. Viewing the HP/Leser filter as a desirable filter, a smooth transition low-pass filter, the optimal finite sample implementation then suggests re-weighting the filter weights by the auto-covariance function of the data or, equivalently, extending the sample with best linear backcasts and forecasts. Viewing the HP/Leser filter as a model, it is clear that output is assumed to be an ARIMA(0,2,2) model, where the output gap is uncorrelated white noise and the potential output growth follows a random walk. Such a model is highly implausible based on economic theory and econometric analysis to be a data generating process for any country’s data. Yet, this is exactly the model that a state-space model (via the Kalman smoother) and the penalized least-squares formulation would use, yielding equivalent results.

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HP filter vs. Modified HP filter – estimate & weights

Citation: IMF Working Papers 2013, 105; 10.5089/9781484399552.001.A001

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Practical News Effect Decomposition for State-Space Models New observations create ‘news effects’ only if they carry some new information, information not predictable from the past data. The representation (31) gives a simple and practical way of calculating and decomposing news effects into components of newly observed data.

The unavailable (or missing) data estimates are simply expected values conditional on the original information set. Padding (filling in) the sample data with these estimates does not change anything, since the information set is identical and there is no new information.

The problem of different sample sizes is easy to convert into a problem of identical sample sizes by padding the data with a model-based forecast and using (29) to carry out the decomposition simply by successive runs of the Kalman smoother.¹¹ The easiest way to see that ‘padding’ the data with conditional expectations or projections does not change the estimates is to realize the structure of the Kalman filter updating step: X_t+1|t+1 = TX_t|t + K(Y_t+1 − Y_t+1|t). In the case of data padded by forecast, the prediction error (information) is zero. The information sets of the original and padded data sets are identical.

This simple and practical approach enables the analyst to investigate a judgement-free forecast of the model and contrast it to actual data. The prediction error is then distributed into the revision of the past unobserved shocks, see e.g. Andrle and others (2009b) for examples using a DSGE model. The decomposition easily accounts for judgement imposed on the filter using a dummy-observations approach.