0306E - Faculty of Social Sciences - Université d'Ottawa

dummy variables. Dummy variables are de…ned by D t;s = 1 if s = t and 0
otherwise. This model, which is labelled as AR; is speci…ed as follows
Á(L)¢ 1 y t = ± 1 D t¡1 + ± 2 D t¡2 + :::± 12 D t¡12 + ² t ; (1)
where Á(L) is a polynomial of order p in the lag operator L with all roots
outside of the unit circle, ¢ 1 is the di¤erencing operator de…ned as ¢ k =
y t ¡ y t¡k for all integers k 6= 0, and ² t » i:i:d: N(0; ¾ 2 ).
Before estimating (1) it is necessary to choose the lag order such that
the residuals are approximately white noise. The order of the AR model
can be selected by conventional methods, such as the Akaike Information
Criterion (AIC) de…ned as AIC = T ln b¾ 2 + 2k; or the Schwarz Information
Criterion (BIC) speci…ed as BIC = T ln b¾ 2 +k ln T; where b¾ 2 = P T
t=1 b² 2 t with
b² t the residuals from the estimated AR(p) model, k = p + 1 the number of
parameters in the model and T is the sample size. The value p 2 f0; 1; :::; pg
that minimizes the AIC or BIC is selected as the appropriate lag order,
subject to passing a Lagrange Multiplier (LM) test for the …rst-to-seventh
order residual autocorrelation at the 5% signi…cance level. The estimation
period for each series is 1985:12 to 2001:02 (182 observations). The last
24 observation are held for out-of-sample forecasting performance. Results
for equation (1) are presented in appendix B (equations B.1, B.2 and B.3).
Diagnostic checks suggest a well …t for the three sectors. R 2 is high, errors
are all normally distributed (except for the transborder sector) and there is
no problem of serial correlation. Further, the RESET test does not imply any
misspeci…cation and the ARCH test suggest no problem for autoregressive
conditional heteroscedasticity.
3.2 An AR(p) Linear Model with Seasonal Unit Roots (SUR)
We label this model SUR. This model reads as follows:
Á(L)¢ 12 y t = ¹ + ² t , (2)
where Á(L) and ² t are as de…ned before. Equation (2) di¤ers from equation
(2) by relaxing the assumption that seasonal patterns are constant over time.
In fact, in equation (2) seasonality is assumed to be stochastic and eliminated
using the seasonal …lter ¢ 12 . Many time series, can be characterized by
seasonal patters which evolve over time, while each season sharing a common
drift ¹. In our domestic series for example this would translate into domestic
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