With the new ivsvar command, you can use instruments instead of short-run constraints to estimate dynamic causal effects. This command is part of StataNow™.

Vector autoregressive (VAR) models describe how a collection of time-series variables interacts. In a VAR model, all variables are endogenous. When we want to estimate dynamic causal effects, we can place theoretical restrictions on the VAR model; these restrictions lead to structural VAR (SVAR) models. Here we focus on short-run SVAR models. Traditionally, these models place restrictions on how shocks impact the endogenous variables. Alternatively, if we have instruments, we can place restrictions on the relationships between shocks and instruments; this allows us to fit instrumental-variables (proxy) SVAR models. In these models, the impact of instrumented shocks (target shocks) on endogenous variables can be freely estimated.

ivsvar estimates the parameters of SVAR models by using instrumental variables. These estimated parameters can be used to trace out dynamic causal effects known as structural impulse–response functions (IRFs). These IRFs describe how a shock to the SVAR model affects the model variables over time.

Let's see it work

One instrument

We have data on outcome variables for industrial production growth (ip_growth), inflation, and interest rate (fedfunds).

. webuse usmacro3
(Federal Reserve Economic Data - St. Louis Fed, 2023-09-01)

. describe

Contains data from https://www.stata-press.com/data/r18/usmacro3.dta
 Observations:           785                  Federal Reserve Economic Data -
                                                St. Louis Fed, 2023-09-01
    Variables:             6                  13 Mar 2024 12:08


Variable      Storage   Display    Value
    name         type    format    label      Variable label                   
 
datem           int     %tm                   Monthly date
ip_growth       float   %9.0g               * Monthly industrial production
                                                growth rate
inflation       float   %9.0g               * Monthly inflation rate
fedfunds        float   %9.0g                 Federal funds rate
oil_inst        float   %9.0g               * Annual net oil price increase
money_inst      double  %9.0g               * Romer and Romer (2004) monetary
                                                shocks
                                            * indicated variables have notes   
 
Sorted by: datem

A VAR model for these three variables includes shocks to industrial production growth, inflation, and the interest rate. We are interested in the effect of an inflation shock. We use an oil price measure, oil_inst, to proxy for the inflation shock. The inflation shock is then called the target shock, and we estimate the effects of this shock. The effects of noninstrumented shocks are not estimated. No restrictions are needed because we have only one target shock.

. ivsvar gmm ip_growth fedfunds (inflation = oil_inst)

Step 1:
Iteration 0:  GMM criterion =  .74206787
Iteration 1:  GMM criterion =  1.602e-31
Iteration 2:  GMM criterion =  2.313e-32

Step 2:
Iteration 0:  GMM criterion =  4.880e-32
Iteration 1:  GMM criterion =  4.880e-32  (backed up)

note: model is exactly identified.

Instrumental-variables SVAR                                Number of obs = 783
VAR sample: 1954m10 thru 2019m12
GMM sample: 1954m10 thru 2019m12

 ( 1)  [e.inflation]inflation = 1


             
 
               Robust                                           
Effect       
 
 Coefficient  std. err.      z    P>|z|     [95% conf. interval]
 
 
 
e.inflation  
 
                                                                
   ip_growth 
 
    -.31198   .4330713    -0.72   0.471    -1.160784    .5368241
    fedfunds 
 
   .0046142    .271441     0.02   0.986    -.5274004    .5366288
   inflation 
 
          1  (constrained)

Note: Underlying VAR fit with 2 lags.
Dependent variables: ip_growth fedfunds inflation
Instrumented shock: inflation
Instruments: oil_inst

There are three coefficients, one each for the impact effect of an inflation shock on industrial production growth, on the interest rate, and on inflation itself. The inflation response is normalized to one, so the other two responses can be interpreted as the relative response to a shock that raises inflation by one unit. On impact, the inflation shock reduces industrial production growth and slightly raises the interest rate. However, we do not have enough evidence to conclude that either impact effect is different from zero.

The dynamic effects trace out the effect of the shock over time. To compute dynamic effects, we compute and graph IRFs.

. irf set ivsvar.irf

. irf create model1

. irf graph sirf, impulse(inflation)

The IRF graph plots the response of each variable to the inflation shock. Eight periods of responses are shown; because these data are monthly, eight periods represents eight months.

Multiple instruments

ivsvar mdist allows multiple instruments to proxy for multiple target shocks. Because there are multiple target shocks, you need to supply some restrictions. You provide Stata with a matrix describing how the instruments are related to the target shocks. Say we have two instruments: an oil price and a monetary policy instrument. We think these instruments are related to the inflation shock and the interest rate shock, respectively. Our theory is

$$ \text{oil}_{\text{instrument}} = p{_1} \times \text{inflation}_{\text{shock}} $$ $$ \text{money}_{\text{instrument}} = p{_2} \times \text{fedfunds}_{\text{shock}} $$

In Stata, we could set up this relationship as

. matrix P = (., 0 \ 0, .)

where the first dot (.) represents parameter \(p_1\) and the second dot represents parameter \(p_2\). By specifying a dot, we instruct ivsvar to estimate the corresponding parameters. And we assume that the two cross-effects, the effect of the interest rate shock on the oil price and the effect of the inflation shock on the monetary policy instrument, are zero.

We estimate the impact effects with

. ivsvar mdist ip_growth (inflation fedfunds = oil_inst money_inst), peq(P)

Estimating short-run parameters:
Iteration 0:  Distance criterion =  .00004274
Iteration 1:  Distance criterion =  .00004117
Iteration 2:  Distance criterion =  .00004117

Refining estimates:
Iteration 0:  Distance criterion =  .00132944
Iteration 1:  Distance criterion =  .00007241
Iteration 2:  Distance criterion =  .00005144
Iteration 3:  Distance criterion =  .00005139

Instrumental-variables SVAR                                Number of obs = 468
Endogenous sample: 1954m10 thru 2019m12
Instrument sample:  1969m1 thru 2007m12

 ( 1)  [e.fedfunds]oil_inst = 0
 ( 2)  [e.inflation]money_inst = 0


Effect       
 
 Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
 
 
 
e.inflation  
 
                                                                
   ip_growth 
 
  -.1212823   .1342488    -0.90   0.366    -.3844051    .1418406
   inflation 
 
   .2240539   .0085336    26.26   0.000     .2073283    .2407794
    fedfunds 
 
   -.020676   .0283709    -0.73   0.466    -.0762819    .0349298

e.fedfunds   
 
                                                                
   ip_growth 
 
   .1604285   .0622792     2.58   0.010     .0383635    .2824934
   inflation 
 
    .025934   .0164775     1.57   0.116    -.0063614    .0582294
    fedfunds 
 
   .4486278   .0149438    30.02   0.000     .4193385    .4779171

e.inflation  
 
                                                                
    oil_inst 
 
   1.298394   .2247589     5.78   0.000     .8578744    1.738913
  money_inst 
 
          0  (constrained)                                      

e.fedfunds   
 
                                                                
    oil_inst 
 
          0  (constrained)                                      
  money_inst 
 
   .1693459     .01252    13.53   0.000     .1448072    .1938846

Wald test of instrument relevance: chi2(6) = 243.5          Prob > chi2 = 0.000
Overidentification test: chi2(1) = .02405                   Prob > chi2 = 0.877
Note: Underlying VAR fit with 2 lags.
Dependent variables: ip_growth inflation fedfunds
Instrumented shocks: inflation fedfunds
Instruments: oil_inst money_inst

The output table is organized by the impact effect of each shock. The first output block displays the impact effect of the inflation shock on the model variables: how an inflation shock affects the industrial production growth, inflation, and interest rate. The impact effects here can be interpreted as 1-standard-deviation shocks. An inflation shock slightly reduces industrial production and increases inflation. Unlike in the previous example, the interest rate now falls slightly on impact rather than rising slightly. But we still do not have enough evidence that the impact effects of the inflation shock on ip_growth and fedfunds are different from zero.

The second block displays how an interest rate shock affects the outcome variables. All three variables appear to rise on impact.

Note that we imposed no restrictions on the impact effects of the shocks on the endogenous variables. All six effects in the first two blocks were freely estimated.

The final two blocks display how each shock affects the two instruments. We have constrained all cross-effects to be zero, so each instrument is affected only by one shock.

If a similar standard short-run SVAR model were fit with svar, constraints would have been placed directly on the impact effects—the first two blocks in the output above. With ivsvar, constraints can be placed on how the shocks affect the instruments, and no constraints are needed on the impact effects.