This tutorial will guide you through the evaluation of dynamic models based on the M1UK data set.
Formulate a model consisting of m on the Constant, m_1, p, p_1, y, y_1, R and R_1, model formulation was explained in the previous chapter. Skip the Model Settings dialog. Move to the Estimate model dialog, accept OLS (ordinary least squares), and set the maximum possible estimation sample, keeping no observations for forecasting. Press OK to send the full-sample estimates to the Results window in GiveWin. This should replicate the model from the previous chapter (§ 3.3). If there are any problems, you can always go back to Chapter 3.
The next major step is to evaluate the estimated model graphically. Select the Graphic Analysis dialog from the Test menu (or press the fourth toolbar button, looking like time-series graphs):
As before, any or all graphs can be saved for later recall, editing and printing.
Two sets of four graphs are offered in this dialog, all with pre-set defaults (to use a non-default sample size, adjust the area in GiveWin).
Accept, and the graphs appear in the PcGets Graphics window in GiveWin.
Click to the second (partially hidden) of these graphs, entitled `Actual and fitted values', as in Figure Figure:4.1. This records the time-series and cross-plots of the actual and fitted values for the whole sample; the residuals scaled by s^, so that values outside the range [-2,+2] suggest outlier problems; and the squared residuals.
Notice that the good fit merely reflects the non-stationarity of m: the high R2 above is consistent with this. Any dependence between successive residuals (reflecting autocorrelation) can be highlighted by editing that panel (usually denoted c) from an index to a line. The graph of the squared residuals suggests there may be some outliers.
Further graphical diagnostic information about the residuals is also plotted, namely their correlogram and partial correlogram, spectrum, histogram, density and cumulative distribution (quantile-quantile, or QQ, plot), as shown in Figure Figure:4.2. The correlogram plots the correlations between successive lagged residuals (that is, the correlation of e^t with e^t-1, then with e^t-2, e^t-3 and so on up to e^t-12). Straight lines are drawn at ±2 s^ to represent approximate 95%confidence intervals. A white-noise (independent) residual would have most autcorrelations close to zero: thus, there is significant dependence between successive residuals here. The partial correlogram plots the partial autocorrelations, so each successive term rn measures the correlation net of lower-order links (i.e., n-1,...,1).
The spectral density is a weighted sum of the autocorrelations, which decomposes the series into components with different frequencies and amplitudes. The spectrum provides a graphical representation, symmetric around 0, and graphed over [0,p] (the horizontal axis in the PcGets graph is scaled by p, reported as [0,1]). A white-noise series has a `flat' spectrum, so we see further evidence of residual autocorrelation.
The histogram is shown with a non-parametric interpolated density estimate (see Ch. 14.5.1.2) and a matching normal: the upper tail shows a clear departure, consistent with the presence of outliers. Finally, the QQ plot shows the transformed cumulative distribution of the sample residuals with that for the normal (which is a straight line): the departure in the upper tail is very visible.
Next, the toolbar button with the box on wheels leads to the dynamic analysis (also on the Test menu):
Dynamic analysis commences with a tabular presentation of the lag structure, the final column of which reports the long-run solution. Chapter 15 provides an explanation. The solved long-run model (or static solution) is calculated, together with the relevant standard errors as follows. Write a single-variable dynamic equation as:
| b0( L) yt=b1( L) xt+et, |
where L is the lag operator so that Lxt=xt-1 and b1(L)=åi=0nb1,iLi is a scalar polynomial in L of order n, the longest lag length. Similarly, b0(L)=åi=0nb0,iLi , with b0,0=1. Let b0(1)=åi=0nb0,i (that is, b0(L) evaluated at L=1), then if b0(1)¹0, the long run is:
| y= |
| x=kx. |
Under stationarity (or cointegration inducing a stationary linear relation), standard errors for derived coefficients like k can be calculated from those of b0(.) and b1(.) . The long-run coefficients are quite well determined, rejecting the null that any is zero.
Dynamic analysis
Lag structure
Lag 0 Lag 1 Sum LongRun
m 0 0.9082 0.9082 1.0000
SE 0 0.0216 0.0216
p 0.2805 -0.1969 0.0836 0.9112
SE 0.1651 0.1661 0.0186 0.0861
y -0.0061 0.1216 0.1155 1.2588
SE 0.1059 0.1083 0.0363 0.3441
R -0.5347 -0.0241 -0.5588 -6.0898
SE 0.1113 0.1226 0.0773 1.1625
Constant -1.1799 0 -1.1799 -12.8577
SE 0.3296 0 0.3296 3.2379
Roots of the autoregressive lag polynomial
real
0.9082
Roots of the distributed lag polynomial in p
real
-0.7019
Roots of the distributed lag polynomial in y
real
-20.0212
Roots of the distributed lag polynomial in R
real
0.0452
Finally, the roots lk,i of the lag polynomials bk( z-1 ) are calculated: values of |l0,i| less than unity are consistent with a stationary or cointegrated relationship.
Access to forecasts is via the Test menu, forecast, or the sixth icon (blue graph with red error bars).
The following screen should be showing:
The available options are to set the forecast horizon (which can be outside the data sample for autoregressive models); the forecast type, namely static (1-step) or dynamic (multi-step, which are really only valid for autoregressive models); the mode of reporting (both tabular in the GiveWin Results window and graphical, as well as whether cumulated forecasts are to be calculated, which could be useful when the regressand is a first difference); the number of pre-forecast observations to graph (the default is the whole sample); and the form of forecast standard errors to be graphed, namely, none, fans, bands or bars. Many of these can be edited in GiveWin if different choices are desired. The maximum horizon is the termination date of the database; and the number of pre-forecast observations depends on the number of forecasts requested during the Estimate Model dialog.
Figure Figure:4.3 shows the forecasts and outcomes for the model used in the next chapter, where the dependent variable is m-p. Error fans, bands, bars, and no standard errors are all illustrated to contrast the possible presentations.
The tabular report is:
Static forecasts from 1987(1) to 1989(1)
step Forecast SE Actual Error t-value
1987-1 1 0.22035 0.01394 0.23683 0.01648 1.183
1987-2 2 0.27132 0.01394 0.28624 0.01493 1.071
1987-3 3 0.31126 0.01394 0.31112 -0.00014 -0.010
1987-4 4 0.34224 0.01394 0.34252 0.00028 0.020
1988-1 5 0.37375 0.01394 0.38719 0.01344 0.964
1988-2 6 0.41239 0.01394 0.41776 0.00537 0.385
1988-3 7 0.42304 0.01394 0.42196 -0.00108 -0.077
1988-4 8 0.43236 0.01394 0.42382 -0.00854 -0.613
1989-1 9 0.43975 0.01394 0.46047 0.02071 1.486
A collinearity analysis is available from the Test menu:
The second-moment matrix MXX=T-1X'X, and the matrix of correlations:
| rij= |
| , |
are reported, together with the eigenvalues of the former, which should all be real and positive. A large ratio of the biggest to the smallest eigenvalue can suggest possible problems, but it must be stressed that measures of collinearity are not invariant under linear transformations, whereas linear models are. Here, we omit the second-moment matrix, but note the correlation matrix, the average entry in which exceeds 0.9:
General model: Wij/sqrt(Wii*Wjj)
Constant m_1 p p_1 y y_1 R R_1
Constant 1.000 0.998 0.997 0.997 1.000 1.000 0.929 0.930
m_1 0.998 1.000 1.000 1.000 0.998 0.998 0.932 0.934
p 0.997 1.000 1.000 1.000 0.998 0.998 0.935 0.938
p_1 0.997 1.000 1.000 1.000 0.998 0.998 0.935 0.937
y 1.000 0.998 0.998 0.998 1.000 1.000 0.930 0.932
y_1 1.000 0.998 0.998 0.998 1.000 1.000 0.930 0.932
R 0.929 0.932 0.935 0.935 0.930 0.930 1.000 0.990
R_1 0.930 0.934 0.938 0.937 0.932 0.932 0.990 1.000
Eigenvalues
real
7.7932
0.1912
0.0104
0.0052
0.0000
0.0000
0.0000
0.0000
Consistent with the very high correlations, the smallest four eigenvalues are reported as zero to four decimal places. Both properties derive from the non-stationarity of the levels variables (often denoted by I(1), for `integrated once', and so needing differencing or cointegration transformations to reduce to I(0)). The Gets approach recommends transforming variables to near orthogonality, which entails an almost diagonal estimated-coefficient covariance matrix. Despite such apparent difficulties, the first round of model simplification in Gets is often best performed on the original levels, whereas the finally-estimated model should be transformed. Thus, we briefly comment on the latter.
All these results together suggest transforming the original model to a more orthogonal representation, in terms of differences and the deviations generated by the static solution (or equilibrium correction). Return to GiveWin, and access the Calculator or Algebra to compute the first differences of the four variables, and the static solution (also available in M1UK.alg, where it is called Eqcms):
| e=m-p-y+6*R+11; |
which is adjusted for its sample mean. The resulting model is:
|
Thus, we recommence from the restricted GUM comprising: Dm, Dp, Dy, DR, their first
lags, and e1 as shown in the following model formulation screen capture, where the current-dated Eqcms is highlighted ready for deletion (otherwise you will get a perfect
fit!):
On re-estimation, the fit has changed markedly as judged by R2, but not s^: both have fallen. Here we focus on the revised collinearity analysis which delivers (the column for the Constant has been omitted):
General model: Wij/sqrt(Wii*Wjj)
Dm_1 Dp Dp_1 Dy Dy_1 DR DR_1 Eqcms_1
Constant 0.763 0.824 0.822 0.428 0.430 -0.012 -0.010 -0.047
Dm_1 1.000 0.621 0.617 0.415 0.327 0.040 -0.205 -0.473
Dp 0.621 1.000 0.943 0.220 0.258 0.072 0.074 0.112
Dp_1 0.617 0.943 1.000 0.216 0.215 -0.046 0.072 0.102
Dy 0.415 0.220 0.216 1.000 0.092 0.098 -0.047 -0.171
Dy_1 0.327 0.258 0.215 0.092 1.000 0.159 0.099 -0.142
DR 0.040 0.072 -0.046 0.098 0.159 1.000 0.189 -0.002
DR_1 -0.205 0.074 0.072 -0.047 0.099 0.189 1.000 0.415
Eqcms_1 -0.473 0.112 0.102 -0.171 -0.142 -0.002 0.415 1.000
Eigenvalues
real
3.6349
1.6813
1.2039
0.8914
0.7307
0.5294
0.1829
0.0967
0.0489
Few correlations are large, and the smallest eigenvalue is 0.05.
Although the mis-specification test outcomes above were not fully satisfactory, to continue with the tutorial we now consider some specification tests.
There are three specification tests: Exclusion restrictions, Linear restrictions and Omitted variables as shown in the Test menu capture above.
Click on this entry in the Test menu to see:
Thus, we are testing the joint significance of Dp_1, Dy and DR_1, which yields:
Test for excluding: [0] = Dp_1 [1] = Dy [2] = DR_1 Subset Chi^2(3) = 3.39251 [0.3350]
These three regressors are therefore jointly insignificant, and could be removed from the model without significant loss of information.
Now, we will test whether the coefficients of Dm_1, Dp and Dp_1 add to unity, so the model has a `real money' growth-rate representation dependent on changes in inflation only, since:
|
|
In (eq:4.1), this hypothesis corresponds to b1+b2+b3=1. Such restrictions are entered in the form:
| Rb=r, |
where b is the k×1 coefficient vector, R the s×k restrictions matrix when s restrictions are imposed (s=1 here), and r is the desired value of the outcome, in an s×1 vector.
Click on the Linear restrictions entry in the Test menu to see:
Implementing the test yields:
Test for linear restrictions (Rb=r):
R matrix
Constant Dm_1 Dp Dp_1 Dy Dy_1 DR
0.00000 1.0000 1.0000 1.0000 0.00000 0.00000 0.00000
DR_1 Eqcms_1
0.00000 0.00000
r vector
1.0000
LinRes Chi^2(1) = 37.2294 [0.0000] **
This restriction is strongly rejected, and could not be imposed on the model without significant loss of information.
This final specification test entry in the Test menu computes LM tests of the significance of the variables dropped during model selection: see §5.5.3.
Although this is an item on the Test menu, we start with a detailed consideration of the involved estimation methods.
Chapter 10 explains the algebra of recursive estimation; the logic is simply to fit the model to an initial sample of M-1>k observations for k regressors, then estimate the equation from samples of size M, M+1, ..., up to T. The output comprises graphs of coefficients, s^ etc., over the sample, as well as checks on parameter constancy.
Formulate a model consisting of mp=m-p on the Constant, mp_1, Dp, R and y_1. Skip the Model Settings dialog. Move to the Estimate model dialog, accept OLS (ordinary least squares), and set the maximum possible estimation sample, keeping no observations for forecasting. Press OK to send the full-sample estimates to the Results window in GiveWin.
Recursive estimation is accessed via the Test menu or pressing on the fifth icon (red graph with a blue line), and leads to the dialog overleaf.
Set the Initial number of observations (M-1 above) to 10 as shown: press on the underlined text, and type the new number in the highlighted box. Next, mark the box Show estimates, t-values and test statistics in separate windows (the default is all output in a single graph). Press the OK button to estimate. In a trice, all T-M estimates and associated statistics are computed: these are denoted by the acronym RLS for recursive least squares.
Recursive instrumental variables simply combines these and will not be discussed explicitly.
Three graphs are produced, provisionally named PcGets: Recursive analysis I, II, III. We consider these in turn, beginning with I, which records the recursive estimates.
First, most coefficients seem constant, but clearly significant only after
the oil crisis (which is still just a sample size of about 40).
The graph of the coefficient of mp_1 over the sample in Figure Figure:4.4b shows that before about 1970, b^1,t was very uncertain, with confidence bands including the origin (so the null hypothesis could not be rejected), and an apparent upward drift in the estimate (although it remains within pre-existing bands).
Secondly, these comments are corroborated by the graphs of the t-values in Figure Figure:4.5, all of which gradually diverge, as should occur with relevant regressors.
Finally, the third graph in Figure Figure:4.6 reports RLS constancy statistics. Panel a shows the RSS at each observation, and panel b, the 1-step residuals, defined by u~t=yt-xt'b^t which are plotted with ±2s^t bands shown on either side of zero. Thus, any u~t which lie outside of the error bands are either outliers, or are associated with changes in s^. There is a steady increase in s^t till around 1975, after which the bands are constant. Next, in panel c, the break-point Chow test (where the forecast horizon H is decreasing from left to right) graphs the actual value of the statistic, whereas the final panel shows its probability value, with the 5%critical value shown as a straight line.
Any of these graphs can be edited and printed, or saved for later recall and printing.
Next, select the Graphic analysis dialog and compare the 1-step with the full-sample residuals:
| ût=yt-xt'b^T |
where b^T is the usual full-sample OLS estimate. The full-sample estimates smooth the outliers evident in the recursive figures, so now the largest is not much more than 3.5 standard errors (partly because s^ increased slightly over the sample). Figure Figure:4.7 records the match.
RLS can be computed backwards as well: essentially the order is reversed -- so earlier, rather than later, observations are added sequentially. Finally, sequential -- or moving window -- estimation can be selected, where the `window' is the number in the Initial number of observations box, so it should not be set too small.
This concludes our tutorial on estimation evaluation. Doubtless you can manage model formulation, estimation and evaluation on your own now: we move on to the really novel aspect of PcGets, namely its model selection procedures.
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