3.1. The Model and Identification of Shocks
The
$k$-dimensional vector
${y}_{t}=\left({g}_{t},gd{p}_{t},{p}_{t},t{r}_{t},{r}_{t},e{x}_{t}\right)$ of quarterly time series includes six variables: government spending (
${g}_{t}$), gross domestic product (
$gd{p}_{t}$), inflation rate (
${p}_{t}$), government revenues (
$t{r}_{t}$), monetary policy rate (
${r}_{t}$) and exchange rate (
$e{x}_{t}$). Exact definitions of the variables are presented in the next subsection. We included both government spending and revenues into the model to cover all possible fiscal policy tools able to curb cyclical fluctuations. The remaining four time series are traditionally considered a basic set of variables necessary for a description of monetary policy effects in a small open economy [
17]. We limit our attention to this basic (but fully sufficient) set of variables due to a large number of parameters in the VAR model.
Our basic VAR has the following form:
in which
${A}_{i}$ are
$kxk$ matrices of coefficients,
$l$ is the number of lags and
${e}_{t}$ are reduced-form random errors with variance-covariance matrix
${\mathsf{\Sigma}}_{e}$. Note that a vector of constants is left out from the specification as we only focus on the cyclical part of the series which has zero mean (see below).
Specification and estimation of the reduced form (1) is generally considered unproblematic within the context of monetary and/or fiscal policy analysis. Our ambition here is to demonstrate that such a view is unwarranted and proper care needs to be paid to this phase of the modelling process.
In applied research, variables usually enter the model (1) either in (log) levels or in first differences. This practice also holds for empirical studies analysing the Czech economy. Although from the statistical point of view, both approaches are possible [
22], their use is conceptually flawed in our context [
2].
Let us summarize the main concerns. First, if the model is estimated in levels, one cannot define any steady-state towards which the system would converge after a shock. Under these conditions, shocks may have explosive nature and do not fade away even after many periods. Permanent nature of such shocks (say monetary policy shock), however, is economically implausible.
The lack of clearly defined steady-state is also crucial in situations when the steady-state serves as a policy target of the authorities. As demonstrated by Andrle and Brůha [
2], the price level cannot be included in the VAR model if inflation-targeting countries are considered since the central bank does not target the level. Application of such a model would lead to false conclusions.
Another problem is that the trend and cyclical fluctuations around it are generally driven by different economic forces. While the variables at business cycle frequencies might echo the effects of implemented policies, trends in the Czech economy do mirror different phenomena such as gradual changes in the inflation target or economic convergence. For example, cyclical fluctuations in the real exchange rate have dramatically different implications for inflation, economic activity and interest rates than its trend which mainly reflect differences in productivity. Similarly, changes in inflation resulting from the time-varying inflation target cannot be attributed to cyclical fluctuations in the domestic product or interest rates. If trends and cycles are estimated within one model by the same set of parameters, the resulting estimates of policy reaction functions are severely biased and commonly exhibit unintuitive economic features such as price puzzle (the price puzzle is always a sign of poorly specified variables in the reduced-form VAR).
The estimation of the monetary-fiscal policy VAR model on first differences is also undesirable. It should be stressed that first differences represent a filtering technique (application of linear filter) which amplifies information on short-term frequencies and attenuates other frequencies. From the perspective of macroeconomic analysis, short-term frequencies are uninteresting and the main focus is placed on business-cycle frequencies which are, however, attenuated by the differencing operation. Moreover, observed macroeconomic trends seem to be too complex to be successfully removed by using first differences. Fox example, a change in the inflation target might again be falsely attributed to the developments in other variables in the model which would lead to a bias in parameter estimates.
The proposed solution to this issue is to decompose all series into a trend and cycle and use a different set of parameters for both components. Since long-term movements in variables are not of the main interest in the context of monetary-fiscal policy analysis, we do not model the trend explicitly and remove it by applying the band-pass filter [
23]. More concretely, we filter out the frequencies longer than 32 quarters (8 years) which are unlikely to be related to policy actions aimed at curbing business cycle fluctuations. To our best knowledge, this approach is unique in the Czech conditions and represents a novel way of analysing interactions between policies adopted by the Czech National Bank and central government.
Once the reduced form (1) has been properly specified, we proceed with the identification of a set of economic shocks and analyse policy reactions to these shocks. Identification process needs to rely on additional theory-based assumptions because it is impossible to estimate structural shocks purely from data. As such, this process may always be subject to controversy as there are many possible ways to obtain responses to shocks from the estimated model. In this paper, we rely on recursive identification scheme and Cholesky decomposition of the variance-covariance matrix ${\mathsf{\Sigma}}_{e}$.
Since recursive identification depends on the ordering of the variables, economic theory needs to be used to support this choice. Our preferred ordering in the pre-crisis period is
${y}_{t}=\left({g}_{t},hd{p}_{t},{p}_{t},t{r}_{t},{r}_{t},e{x}_{t}\right)$ which is also the most commonly used ordering in the existing literature [
7,
10]. As a robustness check we also estimated the model using alternative orderings, however, this had little effect on the results. Slight change in the ordering of variables needs to be adopted for the post-crisis period to account for the fact that the monetary policy interest rate hit the lower bound quickly after the crisis and remained there ever since. The central bank could no longer react to worsening economic conditions via additional changes in the interest rate which implies that the monetary policy rate became the “least endogenous” variable in the model. This situation is therefore better captured by the ordering
${y}_{t}=\left({r}_{t},{g}_{t},hd{p}_{t},{p}_{t},t{r}_{t},e{x}_{t}\right)$.
Although recursive identification is popular, it still might be considered a rough approximation of the reality. A conceptually different approach to shock identification, which is not sensitive to the ordering of variables, consists in specifying economically-motivated sign restrictions on (usually) contemporaneous relations between the variables [
24,
25]. Although we do not make direct use of this approach during the estimation process, we compare our results with economically-founded sign restrictions as a robustness exercise. In practice, we check ex post that the responses to shocks coincide in sign with the assumed co-movements between variables. When the estimated response is not in line with imposed sign restrictions, we refrain from interpreting this specific shock and consider it poorly identified. As a reference guide, we use definitions of shocks proposed in Gerba and Hauzenberger [
12]. These are summarized in
Table 1.
Although sign restrictions presented in
Table 1 are (strictly speaking) only imposed for contemporaneous relations between variables, we interpret them in broader terms—as a theory-implied guideline for the overall path of the response.
3.2. Data
Following earlier literature [
16,
19], we use quarterly data covering a time span from Q1 1999 to Q4 2015. While all data series are available for a given period, exploitation of data prior the year 1999 is problematic because of poor data quality. The first half of the decade was marked by substantial changes in the Czech economy due to the transition from command to free market economy. This period includes a number of one-off shocks that may blur systematic pattern in the data. An additional argument for choosing the year 1999 as our starting point is the adoption of the new monetary policy regime—inflation targeting—during 1998. Our sample, therefore, only contains a period where the central bank applies consistent policy to economic shocks which facilitates interpretation of the results.
To assess changes in mutual interactions between monetary and fiscal policy across time, the sample is split into two subsamples and the model is estimated separately for each of them. The subsamples can be labelled as “pre-crisis” and “post-crisis” period, respectively. The borderline between the subsamples is the year 2009. In fact, the year 2009 which coincides with a sizeable economic downturn of the Czech economy is completely left out from the analysis. This decision is driven by the fact that the crisis was caused by the external environment which is not modelled explicitly. As such, it would have a very undesirable impact on estimated reaction functions. Moreover, from the purely statistical point of view, the crisis year represents an “outlier” that violates the traditional assumptions on the error term of the model.
We note that both subsamples are quite short (in particular the “post-crisis” subsample) and estimation of the model would not be possible without recourse to informative prior information (see next subsection). We do not follow time-varying parameter VAR methodology which is commonly applied in similar settings. There are two main reasons for that. First, this approach can be too susceptible to the noise present in the data which might lead to false changes in parameters. Second and more importantly, we need to adopt specific prior restrictions on parameters which must hold for every quarter of the post-crisis period. These restrictions should account for the zero-lower bound problem and the fact that the Czech National Bank decided to use the exchange rate as an additional monetary policy tool (one-sided commitment to maintain the exchange rate close to the level of CZK 27 to the euro). It would be complicated to adopt such restrictions in a time-varying parameter framework as priors for one period can be overridden by data via Kalman filter recursions.
All data were downloaded from the Czech statistical office and Czech National Bank (ARAD database). All series were seasonally adjusted prior to the analysis. If the seasonally-adjusted series was not available directly from the source, X12 ARIMA procedure was applied. For reasons explained above, the time series were detrended using a band-pass filter proposed by Christiano and Fitzgerald [
23] and only frequencies shorter than 8 years were retained. The only exception is the series of inflation rates where long-term trend is given directly by the inflation target.
Variables entering the VAR model are defined in the following way:
Government spending (in real terms): In line with earlier literature [
8,
19] we define government spending as a sum of government gross fixed capital formation and final consumption. This implies that automatic stabilizers are not considered in the analysis and only purposeful actions to counteract cyclical fluctuations are taken into account. Log transformation was taken prior to the application of the Christiano-Fitzgerald filter. The series was adjusted for the one-off reclassification of institutional unit (SŽDC s.o.) into the central government sector in Q1 2003.
Government revenues (in real terms): We again follow recommendations based on earlier literature and define revenues in a narrower sense. In practice, government revenues are set equal to net taxes (taxes plus social security contributions minus net transfers). The series was transformed into logs prior to the application of the Christiano-Fitzgerald filter.
Gross domestic product (in real terms): Officially published series taken in logs and detrended by the Christiano-Fitzgerald filter.
Inflation rate: We define inflation as a year-on-year change in modified CPI where items related to administered prices are excluded from the consumer basket and the price index is adjusted for the first round effects of indirect taxes (this corresponds to net inflation). This definition may help better describe actual monetary policy actions because the CNB traditionally disregards the impact of indirect taxes on price level in its decision-making process.
Monetary policy interest rate: Although the main monetary policy tool of the CNB is a 2W REPO rate, this rate is routinely replaced by 3M PRIBOR (interbank rate) in empirical research due to the discontinuous character of the former. This discontinuity causes problems for the estimation of parameters. Quarterly averages of the 3M PRIBOR are used in the analysis and the time series is filtered by the Christiano-Fitzgerald filter (no log transformation).
Exchange rate: In our analysis, we make use of real effective exchange rate where upward shifts denote appreciation of Czech koruna. The series was transformed into logs prior to the application of the Christiano-Fitzgerald filter. However, the trend for quarters coming after Q3 2013 was fixed to the value attained in Q3 2013 to account for the exchange rate commitment adopted by the CNB in November 2013.
3.3. Parameter Priors and Model Estimation
Formulation of prior views is technically based on independent Normal-Inverse Wishart prior [
26]. The specific formulation of priors for the pre-crisis and post-crisis period, however, differ. This particularly relates to the autoregressive parameters contained in matrix
${A}_{i}$. For the pre-crisis period, we use a modification of traditional Minnesota prior [
3]. Changes to this system of priors are small but important. Let us recall that original Minnesota prior uses normal distribution for autoregressive parameters where the parameters are centred around zero except for the elements lying on the main diagonal of the matrix
${A}_{1}.$ These are set to 1 if the series is believed to be a random walk or to 0.95 if an AR(1) process is believed to be a better representation of the reality. Variances of normal priors are set according to the following rules [
27]:
where
$i$ refers to the dependent variable in
$i\mathrm{th}$ equation and
$j$ denotes independent variables in that equation. Symbols
${\sigma}_{i}$ and
${\sigma}_{j}$ represent variances of the error terms arising from univariate AR regressions applied to all variables in the VAR model,
$l$ is the lag length and parameters
${\lambda}_{1},{\lambda}_{2}$ and
${\lambda}_{3}$ control the tightness of the prior. We set these parameters to:
${\lambda}_{1}=0.2$;
${\lambda}_{2}=0.5$ and
${\lambda}_{3}=1$ as recommended in the literature [
27].
Our modifications to the original Minnesota prior can be described as follows. Priors for all parameters are defined as before, except for elements lying on the main diagonal of parametric matrices
${A}_{1}$ and
${A}_{2}.$ These are set to 1.1 and −0.4, respectively, and prior variance is set to
${\left(\frac{{\lambda}_{1}}{{1}^{{\lambda}_{3}}}\right)}^{2}$ for both the first and the second lag. The rationale behind this modification is the following: Since long-term trends from all series were filtered out before the analysis, the variability of the transformed macroeconomic series is mainly driven by business-cycle frequencies. This follows from the very nature of the underlying phenomenon these series try to capture. It then seems natural to use this information on the cyclical nature of the series in order to obtain reasonable parameter estimates and this is exactly what we do. Setting prior values on parameters of an AR(2) process equal to 1.1 and −0.4 corresponds to a belief that the process is dominated by business-cycle oscillations (see for details [
28]). Prior variances are chosen so as to roughly cover a range of values leading to similar frequency characteristics of the process. Outside the VAR-modelling literature, priors with identical motivation were already used by Jarociński and Lenza [
29] who tried to elicit reasonable priors about parameters of the AR(2) process when modelling an output gap. However, we are yet unaware of its use in the VAR context.
Specification for the pre-crisis period is completed by a formulation of priors for variance-covariance matrix of the error terms. Since no specific information is available, we use a completely uninformative prior and set the scale matrix to unity matrix and use one degree of freedom.
For the post-crisis period, a different approach is adopted. In this case, more informative priors are needed because of extremely short time series available. Formulation of informative priors in this case is based on elementary principles of Bayesian statistics. In particular, we rely on the fact that posterior distribution may become a new prior distribution of parameters before some new data arrives. This means that we take estimates from the pre-crisis period as a useful starting point for the specification of priors describing the post-crisis period. However, we inflate prior uncertainty (increase variances of the former posterior distributions by a factor of 1.5) in order to give data a greater chance to speak. If posterior distribution remains the same after arrival of the “post-crisis” data, then it can be interpreted as little evidence in data on changes in the behaviour of monetary and fiscal authority after the crisis. Conversely, if the data lead to a significant update of prior distribution (at least in some aspects), then there is a signal of change.
On the top of the approach described above, we need to make one additional step to account for the zero lower-bound problem which occurs when policy rates cannot decrease any further (into a negative territory) to ease monetary conditions. Although the rates were not at zero bound for the whole post-crisis period, they decreased quickly to very low levels and remained on zero level for the large part of the period. For this reason, a prior view that restricts the behaviour of the model in this way is an acceptable simplification of the reality.
As described earlier, we changed causal ordering of the variables to ${y}_{t}=\left({r}_{t},{g}_{t},hd{p}_{t},{p}_{t},t{r}_{t},e{x}_{t}\right)$, but it only eliminates contemporaneous reaction of the interest rate to other variables. To account for the zero lower bound, we also need to eliminate its reactions to other variables that occur with a time lag. This feature can be modelled by setting all parameters in ${A}_{i}$ measuring the reaction of interest rate to other variables in the interest-rate equation to zero. Technically, we set prior mean of the parameters in question to zero and use the prior variance very close to zero (${10}^{-6}$).
Priors for the variance-covariance matrix of the error terms in the post-crisis period are more informative than before, but still loose enough to let the data speak. We set prior scale matrix to the posterior mean of individual elements observed in the pre-crisis period and increased the degrees to freedom to 5.
Computation of posterior distributions is based on traditional Gibbs sampler. Its implementation in the R package BMR [
30] was used to obtain the results. In total, 15.000 of posterior draws were generated from the posterior distribution and first 10.000 draws were discarded as a burn-in period. Retained draws are then used to calculate the posterior distribution of impulse-response (IR) functions. Economic assessment is based on median IR function and 68% credible interval. This interval width is preferred in the case of IR functions [
26] due to potentially highly asymmetric shape of the IR distribution.
The number of lags in the VAR model was set to two. This is in line with the majority of relevant empirical literature (see for example [
6]), but it is also supported by economic reasoning and statistical criteria (i.e., highest marginal likelihood). We recall that autoregressive processes of order two are traditionally used for modelling business cycle fluctuations because their statistical properties suit well to these needs [
31]. This is also reflected in the form of our modified Minnesota prior (see above). Moreover, given a short time span and the number of parameters growing fast with every additional lag included in the model, it would be quite challenging to estimate the VAR with more than two lags since the parameter uncertainty might become unacceptably large.