A Macro-Fiscal Modeling Framework for Forecasting and Policy ...

we want to test the hypothesis that a 1 = 1. Now, under the null hypothesis, the {y t }
sequence is generated by the non-stationary process:
y t = y t-1 + t = y t-2 + t-1 + t = …= y 0 + i=1 t = i=1 to t t (2.7)
Thus, if a 1 = 1, the variance becomes infinitely large as t increases. Under the
null hypothesis, it is inappropriate to use classical statistical methods to estimate and
perform significance tests on the coefficient a 1, which has a non-standard distribution
because variance is infinity. Most conventional asymptotic theories for least-squares
estimation (e.g. the standard proofs of consistency and asymptotic normality of OLS
estimators) assume stationarity of explanatory variables, possibly around a deterministic
trend. Concern for spurious regression is the main reason why the time series analysis is
concerned with stationarity as opposed to conventional econometric theory.
Trend Stationarity
A time series process should be considered trend stationary if after trends are removed,
it is stationary. Phillips and Xiao (1998) define this as follows: if a time series process y t
can be decomposed into the sum of other time series as below, it is trend stationary:
y t = gx t + s t (2.8)
where g is a k-vector of constants, x t is a vector of deterministic trends, and s t is a
stationary time series. Phillips and Xiao (1998) say that x t may be "more complex than a
simple time polynomial. For example, time polynomials with sinusoidal factors and
piecewise time polynomials may be used. The latter corresponds to a class of models
with structural breaks in the deterministic trend."
The process of removing the trend is known as „de-trend‟, which is done simply
by regressing the given series on constant and a trend variable and using the residuals
from that regressions (which is stationery zero) in the subsequent analysis. Alternatively,
the trend variable is included in the regression in which the given series is dependent
variable.
2.3 Structural Breaks
There is now an extensive body of literature in econometrics relating to structural
change. These works highlight the importance of identifying structural breaks in
econometric models, as they lead the changes in the parameters-mean, variance and
trend. If a variable is a trend stationary with structural breaks, then the variable may be
used in its level in the time series analysis, but on the right side of the regression
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