With the new xtvar command, you can now fit a panel-data vector autoregressive (VAR) model to analyze the trajectories of related variables when you observe multiple units or panels over time. This command is part of StataNow™.

VAR models have long been a staple of multivariate time-series analysis, but these models require relatively long series. Now you can apply the same tools to panel data, using observations across panels to compensate for the shorter span typical of these data. You can evaluate the model using moment- and model-selection criteria and Granger causality tests. And you can interpret results using impulse–response functions (IRFs).

Let's see it work

We have data first analyzed by Dahlberg and Johansson (2000) on expenditures, revenues, and government grants for 265 municipalities in Sweden. We want to see how they alter their spending and taxing behavior in response to grants from the central government. We have nine years of annual data for each municipality. Nine observations would be insufficient to fit a time-series VAR model, but because we have observations for a panel of different municipalities, we can perform a panel-data VAR analysis.

Let's load the data and fit a model. We specify the lags(2) option in the xtvar command to include two lags of each dependent variable in each equation:

. webuse swedishgov.dta
(1979-1987 Swedish municipality data)

. xtvar grants revenues expenditures, lags(2)

Panel-data vector autoregression                      Number of obs    = 1,590
Group variable: idcode                                Number of groups =   265
Time variable: year                                   Obs per group:
                                                                   min =     6
Number of moment conditions = 243                                  avg =   6.0
                                                                   max =     6
Fixed-effects transform: FOD
Two-step results
                                (Std. err. adjusted for 265 clusters in idcode)


             
 
              WC robust                                         
             
 
 Coefficient  std. err.      z    P>|z|     [95% conf. interval]
 
 
 
grants       
 
                                                                
      grants 
 
                                                                
         L1. 
 
   .1540956   .0589697     2.61   0.009     .0385171    .2696742
         L2. 
 
   .0639641   .0535182     1.20   0.232    -.0409297    .1688579
             
 
                                                                
    revenues 
 
                                                                
         L1. 
 
  -.0305405   .0161976    -1.89   0.059    -.0622872    .0012062
         L2. 
 
  -.0252658   .0146918    -1.72   0.085    -.0540612    .0035295
             
 
                                                                
expenditures 
 
                                                                
         L1. 
 
   .0142411   .0169733     0.84   0.401     -.019026    .0475082
         L2. 
 
   .0186601    .014656     1.27   0.203    -.0100651    .0473853
 
 
 
revenues     
 
                                                                
      grants 
 
                                                                
         L1. 
 
  -3.558775   .6062701    -5.87   0.000    -4.747043   -2.370508
         L2. 
 
  -1.663785   .2048423    -8.12   0.000    -2.065268   -1.262301
             
 
                                                                
    revenues 
 
                                                                
         L1. 
 
  -.0657256   .0972598    -0.68   0.499    -.2563513    .1249001
         L2. 
 
  -.2419489   .0769968    -3.14   0.002    -.3928599   -.0910379
             
 
                                                                
expenditures 
 
                                                                
         L1. 
 
   .2548482   .0997368     2.56   0.011     .0593676    .4503288
         L2. 
 
   .0240828   .0805262     0.30   0.765    -.1337456    .1819111
 
 
 
expenditures 
 
                                                                
      grants 
 
                                                                
         L1. 
 
  -3.040373   .5854329    -5.19   0.000      -4.1878   -1.892945
         L2. 
 
  -1.526928   .1814479    -8.42   0.000     -1.88256   -1.171297
             
 
                                                                
    revenues 
 
                                                                
         L1. 
 
  -.1236013   .0851185    -1.45   0.146    -.2904304    .0432279
         L2. 
 
  -.2586326   .0759257    -3.41   0.001    -.4074441    -.109821
             
 
                                                                
expenditures 
 
                                                                
         L1. 
 
   .2620421   .0866905     3.02   0.003     .0921319    .4319523
         L2. 
 
  -.0219875   .0767859    -0.29   0.775    -.1724852    .1285101

Hansen's test of overid. restrictions: chi2(225) = 261.92   Prob > chi2 = 0.046
GMM-type instruments: L(2/.).(grants revenues expenditures)

Interpreting the raw coefficients from a panel-data VAR model is not terribly illuminating, but xtvar's postestimation commands make obtaining insights easy. We first perform a Granger causality test to see whether grants Granger-cause expenditures. Granger causality is a rather weak notion of causality; it simply asks whether lagged values of one variable are useful in predicting another variable. To do this, we can use the same vargranger command that works after fitting time-series VAR models:

. vargranger

Granger causality Wald tests

          Equation           Excluded 
   chi2     df Prob > chi2 
            grants           revenues 
  4.1875     2    0.123    
            grants       expenditures 
  1.9027     2    0.386    
            grants                ALL 
  9.6314     4    0.047    
          revenues             grants 
  66.326     2    0.000    
          revenues       expenditures 
  7.4854     2    0.024    
          revenues                ALL 
  85.573     4    0.000    
      expenditures             grants 
  71.603     2    0.000    
      expenditures           revenues 
  11.762     2    0.003    
      expenditures                ALL 
    77.2     4    0.000    

Looking at the equation for expenditures, we see that grants do Granger-cause expenditures. The \(\chi^2\) statistic is 71.6 on 2 degrees of freedom, which provides evidence to reject the null hypothesis of no Granger causality. In other words, if we were to forecast expenditures, we would obtain a lower mean-square prediction error if we included lags of grants in addition to lags of expenditures and lags of revenues.

We can also use the irf suite of commands to obtain IRFs after fitting a panel-data VAR model. We first use irf create to compute IRFs based on the results of our model. We then use the irf graph command to graph the IRFs and evaluate how a shock to each variable is expected to affect the trajectories of each variable in the next eight time periods.

. irf create baseline, set(swedish_govt_irfs)
(file swedish_govt_irfs.irf created)
(file swedish_govt_irfs.irf now active)
(file swedish_govt_irfs.irf updated)

. irf graph irf, byopt(yrescale)

Looking at the plot in the first column of the second row, we see our simple IRF indicates that an unanticipated grant to a municipality leads to lower expenditures in the two years after the grant is received but then returns to the initial spending level.

xtvar also has postestimation tools that allow us to explore the robustness of our findings; these tools are similar to those you would use after fitting a VAR model. For instance, we could investigate the choice of lag length and instruments used to identify the parameters and find an alternative specification using xtvarsoc. And you can check for stability using varstable.