Chapter 10: Exploration of the Output

Introduction

Roots of the Fitted Second-Order Autoregressive Model

Forecasts

Lag 0 Correlation of the Error Terms

The Infinite-Order Representations

Plots of the Impulse Response

Accumulated Effects

Effects of Orthogonal Shocks

Conclusion

Introduction

The output from an application of the VARMAX Procedure includes many elements that all help in understanding the structure of the fitted model. All these elements are indexed in the Results window. See Output 9.1.

Important details include the roots of the fitted models and the coefficients of the infinite-order autoregressive and moving average representations of the estimated model. These numbers are presented in the output as tables and visualized by diagrams. This information forms the basis for the interpretation of the fitted model.

This chapter continues the model from Chapter 9 that was specified and estimated by PROC VARMAX in the final Program 9.6. (This was for the time series of the Danish wages and prices for more than 150 years.) Many tables and plots in the PROC VARMAX output describe the dynamics in the estimated model.

Roots of the Fitted Second-Order Autoregressive Model

In this chapter, the final AR(2) time series of the Danish wages and prices for more than 150 years is studied further. The model was fitted with Program 9.6. The estimated parameter values and the model fit were discussed in Chapter 9. In this chapter, the many other output elements—including tables and graphs—that were generated by Program 9.6 are considered in detail.

Program 10.1 simply repeats the code in Program 9.6.

Program 10.1: Estimating the Parameters of a VARMA(2,0) Model

The output produced by Program 9.6 or Program 10.1 also includes the table of roots for the fitted autoregressive model for the differenced log-transformed price and wage series. Output 10.1 reveals that the polynomial has complex roots, which is the same as saying that the observed series includes some oscillations. The wavelength of the first pair of complex roots is calculated by the reported radian .68, which corresponds to 39 degrees. This frequency corresponds to a cycle length of around 9 years as 2π/.68 = 9.2. The modulus is, however, .55. The cycling behavior of the predictions is damped by a factor of .55, which is the modulus of the root. This factor is raised to the power i, i being the prediction horizon. The second pair of complex roots has no impact because its modulus is very close to 0.

Output 10.1: Roots of the Fitted Autoregressive Polynomial

Forecasts

Figure 10.1 is a forecast plot for the log-transformed wage series without differencing. This plot is also a part of the graphical output produced by Program 10.1. The forecasts simply look like a continuation of the trend and, because of damping, no oscillations are visible. The plot for the price series looks the same.

Figure 10.1: Forecasts of the Log-Transformed Wage Index

However, if the predictions of the series of differences are derived by an analysis of the differenced series DLP and DLW (without further differencing) as in Program 10.2, the periodic structure is just visible for the two series of first-order differences. Figure 10.2 displays the plot for wage series.

Program 10.2: Using PROC VARMAX with the Differenced Series as Input

Figure 10.2: Forecasts of the Differenced Log-Transformed Wage Index

Lag 0 Correlation of the Error Terms

The estimated entries in the covariance matrix for the error terms are shown in the output of Program 9.6 or Program 10.1 (Output 10.2). The corresponding correlation matrix is printed as the first matrix in the table of Cross-Correlations of Residuals (Output 10.3) for these cross-correlations up to lag 2. By default, the table of cross-correlations in the output lists cross-correlations up to lag 12.

Output 10.2: The Variance-Covariance Matrix of the Residuals

Output 10.3: Cross-Correlation Matrix for the Residuals for Lags 0, 1, and 2

Both the wage and the price series are reported as indices, and their variances are of only minor interest. The covariance (COV1_2 = .00149) is significant, having p < .0001. The corresponding correlation .48 (Output 10.3), shows that an immediate interaction exists between the two series. This correlation reveals that shocks to the two series are likely to come simultaneously. In this data set, the time span between two observations is a year. So, coming simultaneously only means that if something happens to one of the series, then the other series reacts within a year. The positive sign of this correlation reflects that the changes in the two series probably are of the same sign. That is, either increasing wages and prices or declining wages and prices are seen in the same year. An increase in one of the series combined with a decline in the other series is seldom seen.

It is a part of the model structure that the cross-correlations for lags different from zero should be zero. This was tested in Chapter 9.

The Infinite-Order Representations

The output also presents the representation as an infinite-order autoregressive model for the series. Of course, this is of no interest in this model because it is itself a finite-order autoregression.

Reactions to a shock in a series later than the same year are reflected in the coefficients of the infinite moving average representation, which was introduced in Chapter 8. By default, these coefficients are printed in the output for lags up to 12. Output 10.4 only presents the coefficients, which are 2 ×2 matrices for lags 1 to 4. The interpretation is that a large input error term, ε_t, at time t, leads to changes in both series in the years to come. In Output 10.4, the coefficients are denoted impulse responses.

The table also gives standard deviations for the estimates so that their significance is easily seen. It turns out that the lag 2 coefficients are of no real importance because only one of the coefficients, .29, for the lag 2 impulse from the price series to the wage series, is substantially larger than twice its standard error. But even if the coefficients are insignificant, the changing signs demonstrate the cyclic part of the model.

Output 10.4: The Moving Average Representation of the Fitted VARMA(2,0) Model

This representation has this form as two individual equations:

ΔX_1t =  ε_1t  + .23ε_1t−1 + .21ε_1t−2 + .11ε_1t−3   + .02ε_1t−4 + .33ε_2t−1 − .02ε_2t−2 − .11ε_2t−3  − .08ε_2t−4  + ...

ΔX_2t = ε_2t  + .19ε_1t−1 + .29ε_1t−2   + .17ε_1t−3 + .06ε_1t−4   + .56ε_2t−1 + .08ε_2t−2 − .08ε_2t−3 − .09ε_2t−4 + ...

Here, ΔX_1t denotes the differenced log-transformed price series DLP, and ΔX_2t denotes the differenced log-transformed price series DLW.

These numbers show that, according to the model, one year of price increases of, say, ε_1t = .1, which is close to 10%, is followed by further increases in prices .23 ×.1. That is, in the next year, there is a further jump of 2.3%, and two years later, there is a jump of .21 ×.1 or 2.1%.

Moreover, the wages will increase by .19 ×.1 or 1.9% the next year. And after two years, .29 ×.1 corresponds to 2.9%. The immediate effect of increasing prices on the wages is not seen by the parameters in this representation. The effect in the same year is modeled by the correlation between the terms ε_1t and ε_2t in the error process. This correlation is positive, .49, and a positive error term ε_1t is probably accompanied by a positive term ε_2t to the wage series. Such a positive term for the wage series would also lead to higher prices and wages due to the mainly positive parameters in the impulse response function.

Plots of the Impulse Response

This representation of the model by an infinite series includes many coefficients as seen in Output 10.4. It is easier to see the structure in the graphs that are produced by PROC VARMAX than to look at the many numbers in Output 10.4. The impulse responses are plotted for increasing lag lengths and can be seen in the graphical output in Figure 10.3 for a push to the wage series and in Figure 10.4 for a push to the price series. These plots more explicitly present the cycling component of the model.

Figure 10.3: Response to Impulse in the Wage Series with Two Standard Errors

Figure 10.4: Response to Impulse in the Price Series with Two Standard Errors

Accumulated Effects

The effects are also cumulative. That is, the total effects on each series are calculated as simply the sum of the effects up to a particular lead value. In the present example, these values describe the total effect to the series when one of them experiences a sudden shock. The numbers are printed in Output 10.5.

Output 10.5: Accumulated Impulse Response

The result is that a shock to the price series (that is, a large value of the error term ε_1t) after, for example, 4 years has a total effect of 1.57 times the immediate effect to the price series. Because the series are log-transformed, this corresponds to a multiplicator effect of 1.57, which means that increasing prices one year lead to more than a 50% rise in prices the following years. In other words: Inflation is persistent!

But a shock to the price series, ε_1t, also affects the wage series. The effect in the same year due to the correlation is not included in the table in Output 10.5, but it is seen that workers after 4 years are compensated 71% by increasing wages.

The plots for the accumulated effects are presented in Figure 10.5 and Figure 10.6. These plots show that the effect of the wage series on the price series looks similar to the effect the other way around.

Figure 10.5: Accumulated Response to Impulse in the Wage Series with Two Standard Errors

Figure 10.6: Accumulated Response to Impulse in the Price Series with Two Standard Errors

Effects of Orthogonal Shocks

A third part of the output presents the effect of an orthogonal shock on one of the series. The idea is that the error term is present in only one of the series, and that it is not at all a part of the error for the other series, as is usually the case because the error terms are correlated. These tables and plots show what happens to both series in the years after a specific event that occurred in just one of the series.

Mathematically, this effect is calculated by an orthogonalization of the correlation matrix Σ of the error terms, ε_t. The covariance matrix is factorized as Σ = PP^T, where the matrix P could be taken as a lower triangular matrix. The error processes is standardized to variance 1 in this representation, and the individual error processes are independent. The orthogonalized impulse response is defined as the coefficients to these orthogonalized errors. In the output series, these coefficients are represented as an infinite series in lagged values of orthogonalized errors.

Output 10.6 gives the numbers, which are all close to zero, the largest value being 0.057. The absolute values of these parameters are all small due to the standardization of the error processes to variance 1. The actual numbers (for instance, 0.057) for lag 0 are for this reason only a consequence of the variances of the two series. Nevertheless, the parameters are significant as measured by their reported standard deviations.

Output 10.6: Orthogonalized Impulse Response

These parameters, including their confidence bands, are shown in Figure 10.7 and Figure 10.8. The effect for further years varies according to the damped periodic structure of the model. This is because of the complex roots of the autoregressive polynomial.

Figure 10.7: Response to a Specific Impulse in the Wage Series

Figure 10.8: Response to a Specific Impulse in the Price Series

The simultaneous effect from the first series (here the price series) on the other series (the wage series) at lag 0, is, by the construction, equal to 0. The parameter for the effect the other way is nonzero because this parameter represents the lag 0 correlation between the series. This would be reversed if the two variables were stated in the opposite order—that is, in the MODEL statement.

Conclusion

In this chapter, we studied the dynamics of the VARMA model for the wage and price series that were estimated in Program 9.6 (Program 10.1). PROC VARMAX offers many tables and graphs for this purpose.

In the procedure output, you find the roots of the autoregressive part of the model. In mathematical terms, these roots show that the series in this example includes some cycling behavior. A periodic structure is also seen in the various impulse response plots. But in the forecast plots, details are hidden behind changing trends and by the noise component in the model.