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Unrestricted Vector Autoregression(VAR)

Guy LionJanuary 2, 2016

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Introduction

There are several different VAR models. Here, I am studying only the most common VAR model: unrestricted VAR.

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Data

I am using a recent data set from H6 Money Aggregate focusing on DDA looking at quarterly change in such DDA outstanding measured as First Difference in Natural Logs(FDNL)* from the early 1960s to 2015 Q2.*Only reason I used FDNL was because of the data transformation I used at the time.

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VariablesEndogenous variables: H6 DDA, Personal Income, 10 Year Treasury. All those variables are on an FDNL or FD basis. Those are the Z variables (no distinction between Y and X variables) that will be “VARed.” This means that the unrestricted VAR model will generate three different multivariate regressions estimating in turn each of the three variables.

Exogenous variables: those are variables that will not be “VARed.” They will not be regressed. Those include seasonality variables (Q1 and Q4) for two different state breaks (Pre 2001 and Post 2001), unlimited FDIC NIB guarantee period in 2009-2010, and Dodd & Frank unlimited guarantee period in 2011-2012.

The above indicates that VAR is very flexible. If we had many independent variables and did not care about “VARing” them, we could have moved them from the Endogenous Group to the Exogenous Group.

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Comparing Multiple Regression vs. VAR

Coeff. SE t Stat P-valueIntercept -0.9% 0.6% -1.46 0.14 Pers. Inc. Lag 2 1.15 0.28 4.14 0.00 10 Yr T Lag 1 -1.66 0.50 -3.33 0.00 Q1 < 2001 -6.5% 0.7% -9.96 0.00 Q4 < 2001 3.8% 0.6% 6.04 0.00 Q1 > 2001 -4.9% 1.0% -4.92 0.00 Q4 > 2001 7.1% 1.0% 7.03 0.00 TAGP 1 & 2 4.1% 1.3% 3.22 0.00 DF 5.7% 1.3% 4.44 0.00

Resi dual st andar d er r or : 0. 03401 on 202 degr ees of f r eedomMul t i pl e R- Squar ed: 0. 6301, Adj ust ed R- squar ed: 0. 6081 F- st at i st i c: 28. 67 on 12 and 202 DF, p- val ue: < 2. 2e- 16

Multiple Regression VARRegression Statistics

R 0.783R Square 0.614Adj. R Square 0.599RMSE 3.43%# 217

For the Multiple Regression, we selected the one best lag for Personal Inc. and 10 Yr. Treas. For VAR, we selected the best cumulative lags up to Lag 4. VAR selected the first two lags. Notice, VAR always includes one or more autoregressors (Y t-1, Y t-2, etc.)

Coeff. S. Error t stat p valueEndogenous h6dda. l 1 -0.08 0.06 -1.24 0.22variables pi . l 1 -0.53 0.32 -1.66 0.10with lag 1 & t en. l 1 -1.25 0.53 -2.35 0.02lag 2 h6dda. l 2 0.03 0.05 0.53 0.60

pi . l 2 1.38 0.31 4.50 0.00t en. l 2 -0.79 0.54 -1.47 0.14

Exogenous const -0.6% 0.01 -0.82 0.41variables q1a -5.7% 0.01 -7.10 0.00with no lags. q4a 4.3% 0.01 6.31 0.00Seasonality. q1b -4.1% 0.01 -3.60 0.00Government q4b 7.2% 0.01 6.95 0.00guarantees t agp 3.9% 0.01 2.82 0.01

df 5.9% 0.01 4.17 0.00

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Multiple Regression vs VAR Model Review• Even though the VAR model has six endogenous variables vs. an equivalent of only two for the

Multiple Regression model, their respective Goodness-of-fit statistics are hardly distinguishable (R Square, Adjusted R Square, Standard Error);

• VAR has several variables that are far from being statistically significant (the two autoregressors: Y t-1 and Y t-2);

• In VAR, Personal income lag 1 has a questionable negative sign from an economic theory standpoint;

• The seasonality variables and the deposit insurance guarantee variables have similar values in both models.

• Forecasting is not going to work well with VAR because of the autoregressors (y t-1, y t-2). You will have to rely on their respective estimates just two and three periods out. The customary way to forecast with VAR (sometimes called dynamic forecasting) entails rerunning your model every period and forecasting one single period at a time. That’s essentially treating medium term forecasts as a consecutive series of single-period forecasts. That’s not true forecasting.

In view of the above, Multiple Regression appears much preferable over VAR for forecasting. This example is representative of the mentioned weaknesses of VAR (too many autoregressors, statistical significance and regression sign issues.

One of the main reasons of using VAR is to measure the long term impact of an immediate shock of one of the endogenous variables on the true dependent variable Y. This is captured within the Impulse Response Function of VAR.

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The main benefit of VAR: Impulse Response Function (IRF)“[An IRF] trace out the response of current and future values of each of the variables to a one unit increase in the current value of one of the VAR errors, assuming that this error returns to zero in the subsequent periods and that all other errors are equal to zero”

James H. Stock and Mark W. Watson, July 2001.

The VAR framework to generate an Impulse Response Function (IRF):1) y = c + coeff(y t-1) + coeff(x t-1) + error term2) x = c + coeff(x t-1) + coeff(y t-1) + error term3) 1 unit increase shock in x (impulse variable) = impact on y (response variable) over next # of periods.

The first two steps are just two simple regressions. The first one regresses y using y t-1 and x t-1 as independent variables. The second one does the same thing but reverses what is the independent variable. So you now regress x using x t-1 and y t-1 as independent variables. The IRF is derived by measuring the impact on y of a one unit upward shock in x captured as the error term of the regression estimating x. This shock occurs in period 0 (or 1 depending on how you name your periods) and is measured over several numbers of periods forward (typically 8, 10 , or 12 if you are dealing with months or quarters).

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IRF: 10 Year Treasury is impulse variable. H6 DDA growth is response variable

The IRF makes some sense and is actually modest when you figure that the regression coefficients for 10 Yr Treasury are - 1.25 for Lag 1 and another -0.80 for Lag 2.

The graph represents the impact of an unexpected upward sudden 1 pct. pt. increase in 10 Year Treasury on H6 DDA growth over the next 10 quarterly periods. Remember, this phenomenon is extracted by capturing the impact on Y of the Error term within the 10 Year Treasury regression.

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IRF: Personal Income is impulse variable. H6 DDA growth is response variable

The graph represents the impact of an unexpected sudden upward 1 % increase in Personal Income over H6 DDA growth over the next 10 quarterly periods. Remember, this phenomenon is extracted by capturing the impact on Y of the Error term within the Personal Income regression.

The questionable zig-zag (negative then positive) impact of Personal Income on H6 DDA growth simply reflects the regression coefficients of Personal Income Lag 1 and Lag 2 (Lag 1 was negative; Lag 2 positive).

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Are IRFs for real?Maybe not… After all, a model Error term is very often just that. It is an Error reflecting that the model is missing variables that would further explain out the residual of the regression and reduce the Error. In other words, the Error may be more Noise than Signal. Professors have advanced this point (Stock & Watson 2001). In such a situation, an IRF function may not be representative at all.

One has to wonder if our IRF with Personal Income as the impulse variable and H6 DDA growth as the response variable is a case in point. Based on economic theory, do we expect an unanticipated 1 percentage point increase (Residual + 1%) in Personal Income to cause a -0.5% decrease in DDA growth a quarter later, than a + 0.6% growth two quarters out, etc. Yet, those movements make some sense when referring to the VAR regression coefficients for the DDA regression with Personal Income lag 1: – 0.53; and Personal Income lag 2: + 1.38. Both those regression coefficients were statistically significant. But, it does not mean the whole regression is well specified. It could be overfit with too many repressors (y t-1 & y t-2) and lagging variables. The latter appears embedded in the structure of VAR models.

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Another interesting VAR output: Forecast Error Variance Decomposition (FEVD)

> f evd( var 3, n. ahead = 10)$h6dda

h6dda Pers. Inc. 10 Yr. Tr. Total1 100.0% 0.0% 0.0% 100.0%2 94.4% 3.1% 2.4% 100.0%3 88.4% 7.8% 3.8% 100.0%4 88.4% 7.8% 3.8% 100.0%5 87.7% 8.6% 3.7% 100.0%6 87.7% 8.6% 3.7% 100.0%7 87.6% 8.6% 3.7% 100.0%8 87.6% 8.7% 3.7% 100.0%9 87.6% 8.7% 3.7% 100.0%

10 87.6% 8.7% 3.7% 100.0%

If I interpret this table correctly (referring to Stock & Watson, 2001, pg. 7, 25), it suggests that Personal Income and 10 Yr. Treasury do not explain any of the DDA growth estimate error in the first quarter (common). But, that they explain together up to 12.4% of such errors in the out quarters of the forecast. And, Personal Income explains a bit more than twice as much as 10 Yr. Treas. Those findings are not

intuitive when referring to either the VAR model or the IRFs. You would have thought those variables explained a lot more of the DDA growth error. But, that’s what I grasp based on the mentioned reference.

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Technical Notes

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Endogenous vs Exogenous VariablesEndogenous and Exogenous variables each need their own different matrix object. The matrix objects can easily be defined from the very same data set (h6dda4). The endogenous matrix3 includes H6 DDA, Personal Income and 10 year Treasury. The exogenous matrix4 includes the seasonality variables and the FDIC and Dodd & Frank NIB guarantee variables. See the R coding and their explanations below:

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R vs. EViewsYou can develop models in EViews as easily as you can in R. You can just as readily generate the IRFs and FEVDs. You get the exact same results generated just as quickly.

EViews at times gives you more options. Such is the case for IRFs where you have several different options on how to measure the impulse (the unit of error). However, when it comes to figuring out the FEVDs, EViews tells you there is only one type of IRF specification that is valid (if you want to generate related FEVDs that sum up to 100%).