17.3 Specifying a VAR Model

The starting point in the Sims methodology (Sims; 1980) is the formulation of an unrestricted VAR model. We will specify a VAR model of order $ p$ (VAR($ p$)) in the following general form:

$\displaystyle y_{t} = \nu + A_{1} y_{t-1} + \ldots + A_{p}y_{t-p} + u_{t} \; , \quad\quad t=1,\dots,T \; ,$ (17.1)

where

To analyze a model in the form (17.1) we need to set the model type to Full VAR after selecting Model type in the Main Menu:

\includegraphics[scale=0.7]{mts_main}\includegraphics[scale=0.7]{mts_model}
In our model $ y_t$ is specified with $ y_t=(\Delta (\ln M),\Delta (\ln Y),I)^T_t$. The variable order in the vector $ y_t$ is given by the data matrix x. The next steps include finding a suitable model order $ p$ and estimating the model parameter matrices $ \nu,A_1,\dots, A_p$. Having done this we should conduct a residual analysis to check the whiteness assumption. If the residual analysis is satisfactory we can use the model for interpretation and forecasting.


17.3.1 Process Order

We can use economic theory or information contained in the data for specifying the model order $ p$. Since we have no a priori knowledge from theory we use statistical tools for choosing an appropriate $ p$. The quantlib multi provides the FPE, AIC, HQ and SC criteria (see Severini and Staniswalis; 1994, Chapter 4). They all compare different VAR($ p$) models with $ p=0,\dots,p_{\max}$ with respect to some objective function. The order $ p$ which optimizes the function is the recommended order.

Before we apply the order selection criteria we must set the highest possible order $ p_{\max}$. This can be difficult: In order to avoid an optimum at the edge and to restrict the parameter space not too much $ p_{\max}$ should be reasonable large. On the other hand $ p_{\max}$ must not be too large since we need at least $ p_{\max}+1$ presample values which reduces the sample size $ T$ and results in unprecise estimates or worst in a model that cannot be estimated. Since we deal with quarterly data we should consider at least the periodicity as a possible process order. Moving a bit ``away'' from the periodicity we set $ p_{\max}=7$.

In the Main Menu we select Model specification and estimation and prepare the subsequent call to the model selection criteria:

\includegraphics[scale=0.5]{mts_main_full}    \includegraphics[scale=0.5]{mts_var_order1}
For that we must divide the data set in a presample and sample period. We can ignore the input fields Order and Mean adjusted (set to 0). The presample period must contain at least $ p_{\max}$ observations. Since we have differenced the data once one more observation is 'lost'. Therefore the Beginning of sample is set to $ p_{\max}+2=9$. If the sample is not split appropriately an error message appears in the output window which indicates the problem.

Press OK to enter the menu of VAR estimation results (main results menu) and select VAR order criteria. Here we are asked to input $ p_{\max}$:

\includegraphics[scale=0.7]{mts_var_main}    \includegraphics[scale=0.7]{mts_var_order2}
The results of the order selection criteria will be presented in a separate window. The optimum for every criterion is found at the minimum value:
\includegraphics[scale=0.6]{mts_var_order3}
The optimum values are FPE $ (4)=-17.9254$, AIC $ (4)=-17.9821$, HQ $ (4)=-17.6276$, and SC $ (1)=-17.3654$. The recommendation of the SC-criterion is quite different from the others. Such a result is not uncommon. For a detailed discussion about the properties of the criteria see Lütkepohl (1993, Chapter 4).

We start our analysis with $ p=4$ but should keep in mind the other possible process order. Thus we start with a VAR(4) which is the most general model supported by the data. This also includes a VAR(1) by setting $ A_2=A_3=A_4=0$.


17.3.2 Model Estimation

In order to estimate a VAR(4) we need to go back to the Main Menu and select Model specification and estimation again. However, this time we set the Order to 4 for estimating a VAR(4) model:

\includegraphics[scale=0.5]{mts_main_full}

\includegraphics[scale=0.5]{mts_var_est1}
Press OK to enter the results main menu. The VAR(4) is estimated by multivariate least squares. Next we view the estimates $ \hat{\nu},\hat{A}_1,\dots , \hat{A}_4$ and their $ t$-values:
\includegraphics[scale=0.8]{mts_var_est2}
It can be seen that not all elements of the parameter matrices are significant different from zero. Especially in $ \hat{A}_2$ and $ \hat{A}_3$ we observe only one significant value. This could be the starting point for choosing a subset VAR where single elements of $ A_i$ are restricted to zero.

Selecting Covariance matrix of residuals from the main results menu displays the estimated residual covariance and correlation matrices.

\includegraphics[scale=0.8]{mts_var_est3}
The correlation matrix tells us something about the contemporaneous correlation structure in the residual vector $ u_t$. We note here that there is no correlation in $ u_t$. At a later point we will come back to the implications of this feature.


17.3.3 Model Validation

In Subsection 17.3.2 we have estimated a VAR(4)-model. Since we did not know the ``correct'' order we used statistical tools to find a reasonable one. Some estimation results were presented. Partly they are based on properties of the estimator (limiting normal distribution) which assume certain conditions. Whether these conditions hold is checked in this subsection. One can think of a residual analysis, tests for nonnormality and tests for structural change. Here we will consider the residual analysis and a test for nonnormality in more detail.

Checking the whiteness of the residuals is a prerequisite for drawing valid conclusions from the $ t$-values presented above. If we want to compute reliable forecast intervals we need to check the normality of the residuals in addition.

From the main results menu we select Residual Analysis which enables us to go through the three steps of residual analysis in multi :

\includegraphics[scale=0.7]{mts_var_main}\includegraphics[scale=0.7]{mts_var_val1}

Individual residual analysis

First we have to select one equation. Then we have the chance to do some transformations to the estimated residuals $ \hat{u}_{i,t}$. We selected here Residuals which leaves $ \hat{u}_{i,t}$ untransformed.

\includegraphics[scale=0.7]{mts_var_val2}\includegraphics[scale=0.7]{mts_var_val3}
From the following menu we present here the Plot of residuals. We do this for all equations.
\includegraphics[scale=0.7]{mts_var_val4}
In the residual plots the unit of measurement is one standard deviation. In other words, the residuals are normalized to have unit variance. Thus, if many residuals exceed 2 in absolute value this may be evidence for nonnormality or nonlinear features that are not adequately captured by the model. Furthermore we might look for distinct patterns in the residual plots that rule out whiteness.
\includegraphics[scale=0.4]{mts_var_res_m}

\includegraphics[scale=0.4]{mts_var_res_y}

\includegraphics[scale=0.4]{mts_var_res_i}

Multivariate portmanteau statistic

Checking the white noise assumption for the residuals is a central issue. Many inferential procedures rely on this assumption.

The menu point Multivariate Portmanteau statistic provides two tools. Here we look at the residual vector $ u_t$ at time points $ t$ and $ t-i$. For these we compute the $ i$-th autocorrelation matrix $ R_u(i)$. White noise means zero autocorrelation for all $ i\ge 1$. Before checking the autocorrelation functions and carrying out an overall test we are asked to input a maximum lag $ h$ we want to check autocorrelation for:

\includegraphics[scale=0.7]{mts_var_val5}
For the overall test to work the maximum lag of the autocorrelations to be included must exceed the order of the process as otherwise negative degrees of freedom of the approximating $ \chi^{2}$ distribution will result. If a lag $ h$ less than or equal to the VAR order is specified a warning is given and the statistic is computed for the smallest feasible lag. Generally, the $ \chi^{2}$ approximation to the true distribution of the modified portmanteau statistic may be inappropriate for small lags $ h$. At the same time we must make sure that $ h<T$ for obvious reasons.

Here we have chosen $ h=20$. The resulting plots of the autocorrelation functions appear. The autocorrelation plots come along with approximate $ \pm 2/\sqrt(T)$ confidence bounds. These plots do not exhibit significant autocorrelations. Especially the $ \hat{\rho}_i,t$ with $ t\le p$ are much smaller than the approximate confidence bound which is a good result since the exact confidence bound for smaller autocorrelation lag can be much smaller than the approximate.

\includegraphics[scale=0.6]{mts_var_acf_all}
A test for the overall significance of the residual autocorrelations up to lag $ h$ appears in a second display. It is the result of the test

$\displaystyle H_0: \textrm{\boldmath$R$}_h=(R_u(1),\dots ,R_u(h))=0 \qquad \textrm{vs.} H_1: \textrm{\boldmath$R$}_h \neq 0$ (17.2)

The value of the modified portmanteau statistic $ \bar{P}_{h}$ (see Lütkepohl; 1993, Chapter 4) is shown:
\includegraphics[scale=0.6]{mts_var_val6}
As expected we cannot reject $ H_0$ at a $ 95\%$ significance level. This result is in line with the shown residual autocorrelation functions above. The residuals of our model do not exhibit autocorrelation.

Multivariate normality test

Multivariate Normality test displays the $ \chi^{2}$-statistics associated with the skewness and kurtosis of the residuals which may be used for tests of nonnormality.

\includegraphics[scale=0.6]{mts_var_val7}
The result shows that normality of the residuals is not rejected on grounds of the test of kurtosis and the joint test of skewness and kurtosis. The implication is that the confidence intervals computed for the forecasts are reliable.