univariate 3.2
Footnote: time series meets regression
squares calculations have been invented. 12.4-5 of U/G Econometrics discusses some of them. Testing the hypothesis that = 0 in the model
The autoregressive model rst appeared in econometrics not in 1970 but around 1950 when the classical regression model was extended to include serially correlated disturbances. Introduction to Econometrics introduced the regression model with autoregressive errors; in its simplest form this is yt = xt + t t = t1 + ut : ut IN(0, 2). See chapter 12 of U/G Econometrics. The model can be estimated by maximum likelihood but this involves the use of numerical optimisation and to avoid this all sorts of tricks involving least
yt = xt + t t = t1 + ut : ut IN(0, 2) inspired the most famous diagnostic test in econometrics, the Durbin-Watson test. See U/GE 12.6.1. The DW test statistic, based on the least squares residuals, can be derived using the LM/score principle. Another LM serial correlation test based on the least squares residuals and given on EViews is the BreuschGodfrey test. This contemplates a more general form of serial correlation, an ARMA(p, q) t 1t1 ... ptp = ut 1ut1... q utq ut IN(0, 2) H0 1 = ... = p = 0 = 1 = ... = q
The null hypothesis of white noise requires that p + q parameters are zero and the test statistic which is based on the autocorrelations of the least squares residuals is distributed as 2 in large samples. p+q The regression model with autoregressive errors specication has fallen from favour because it seems more plausible to include any dynamics in the systematic part of the model, using a model of the form yt = yt1 + xt + xt1 + t : t IN(0, 2) called the autoregressive distributed lag modelch. 15.5. The ADL model is an AR model with exogenous and lagged exogenous variables (xt and xt1) added. Usually there are higher order lags as well. ADL models can be estimated by least squares because the regressors and the (contemporaneous) error term are not correlated. The usual t and F tests are valid only approximately when the sample is large.
3.3
Multivariate time series models
We now consider time series models with more than one variablevector time series, they are called.
There are vector generalisations of the MA and ARMA processes but they are rarely used.
The vector autoregressive model (VAR) is very widely used.
The VAR is the natural extension of the 1-dimensional AR discussed earlier. In the case of 2 variables, a single lag and no intercepts the VAR is yt = 11yt1 + 12xt1 + 1t xt = 21yt1 + 22xt1 + 2t.
The VAR can be interpreted as a special case of the dynamic SEM. It is the reduced form in the case when there are no exogenous variables. When the VAR was rst studied in the 40s it was seen as uninteresting because no economics could be hung on it. But the disadvantages of the 40s became the advantages of the 70/80s. The VAR is not committed to any theoryit does not even divide variables into exogenous and endogenous for all are endogenous. All the examples will involve two variables (hence are bivariate models) but it will be convenient to use matrix notation when discussing general issues. The bi- example can be put in matrix form as zt = Azt1 + t
by making the identications zt = yt xt , A= 11 12 21 22 , t = 1t 2t .
But the matrix equation holds good if the vectors have m components and A is an m m matrix. In empirical work models with only 1st order lags are unusual and so naturally there is a kth order VAR zt = A1zt1 + ... + Ak ztk + t where the A s are square coecient matrices. Notice how the number of parameters grows with the number of variables and the number of lags. With 2 variables and 1 lag each equation has 2 coecients. With 6 variables and 4 lags each equation has 24 coecients. I wont develop the k-th order model in any detail because the interesting points can all be made in terms of the 1st order process.
To complete the specication we need a description of the properties of . We will make the usual assumptions (and also show how they are written in matrix terms). For the expectations, Eit = 0, i = 1, 2 Et = 0 or the expectation of the vector disturbance is the zero vector. For the variances and covariance, var(it) = 2, i = 1, 2; cov(1t, 2t) = 12 i 2 12 1 = . var(t) = 21 2 2 or the variance matrix of the disturbance vector is the matrix . Note that variance matrices are always symmetric because 12 = 21.
Granger Causality
The VAR rst made an impact in connection with Granger causality. Underlying the Cowles division of variables into exogenous and endogenous was a notion of causality: exogenous variables are causes.
The original Cowles econometricians thought that questions of causality were to be settled by economic reasoning not by statistical analysis.
Later econometricians argued that the Cowles notion of causality could be tested and that the exogeneity tests developed in the 70s and 80s (see above) were ways of doing this.
Clive Granger, a non-structural econometrician, took a dierent line. Granger (1969) proposed a data-based time-series method for determining causality.
Usefulness in forecasting is not interpreted in an absolute sense. x is useful if it makes possible better forecasts than would be made using only the past history of y.
During the 70s Christopher Sims applied Grangers ideas to the big question in monetary economics of the dayon the causal relationship, if any, between money and nominal income.
The idea can be formalised as follows. Forecasting methods are often evaluated by mean square forecast error: E(yt+1 yt+1)2. Suppose one forecast is based on ys past history only and the other is based on xs past history as well. Then x is a cause of Y in Grangers sensewhen E (yt+1 yt+1(yt, yt1, ...; xt, xt1...))2 < E (yt+1 yt+1(yt, yt1, ...))2
Grangers notion is very natural in time series forecasting :
x is a cause of y if x is useful for forecasting y. (So this notion of causality ts the notion that causes precede eects but does not apply naturally to contemporaneous x and y.)
Applying these ideas to the bi-var yt = 11yt1 + 12xt1 + 1t xt = 21yt1 + 22xt1 + 2t
the history of x will usually help in forecasting y. But if 12 = 0 the best forecast of yt+1 is 11yt and x does not Granger cause y. Similarly, if 21 = 0, then y does not Granger cause x. It is easy to nd cases where the relationship between x and y is not causal but which nevertheless t the Granger denition. Barometer readings may foretell storms but do not cause them. More generally x may be a leading indicator of Y without necessarily causing y. Passengers on a train move towards the do...