Lecture 4: Persistence, Non-Stationarity and Spurious Regression
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what happens if weak dependence and stationarity do not hold & what to do
assumptions for validity of OLS are not satisfied
need to transform the model to stationary process that satisfies weak dependence
show that AR(1) is nonstationary if θ = 1 and y0 = 0
page 2
random walk model
show random walk shows highly persistent behaviour (value of y today determines the value of y j periods away)
page 2
unit root meaning
page 4
shocks have permanent effects
p=1 in classic AR(1) model
definition of integrated process and order of integration
integrated processes: turn unit root processes into weakly dependent ones
order of integration: number of differences to arrive at weakly dependent
transforming random walk to weakly dependent
page 4
I(0) vs I(1)
page 4
difference stationarity
process becomes stationary after prcess is applied to it (I(1))
show model with deterministic trend is not weakly stationary
page 5
trend stationary
if you remove trend in nonstationary model and it becomes stationary, the variable is trend stationary
page 5
consequences of trend stationary and difference stationary
page 6
testing for unit root
test whether theta is 1 or different to 1
basically testing whether difference stationary or not
page 7
why can’t we use standard t test for unit root
standard t distributions are not applied since OLS regression assumes stationarity (yt is nonstationary)
page 7
MacKinnon tables
used to look up critical values for DF tests - more accurate and provide values for more sample sizes
DF dist has fatter tails - less likely to reject null of unit root when it’s true
Dickey Fuller test of AR(1) model
page 8
3 cases for MacKinnon critical values & when to reject unit root and conclude series is stationary
page 8
if more negative, reject null and most likely stationary
more positive = can’t reject null and may be non stationary
issue with standard DF
page 9
augmented DF test
used if serial correlation in errors
add lagged differences of yt to regression equation:
page 9
how does augmented DF test work
page 9
how do lag terms in augmented DF test remove serial correlation
lagged terms act as additional regressors that capture the patterns in past errors
page 9
consequences of too many lags or too little in ADF?
too few: serial correlation remains - biased test remains
too many: loses power → harder to reject unit root when series all stationary
problem with DF and why (low power)
low power - increases risk of failing to reject null hypothesis even if it’s false
why it has low power - adjusted critical values to reduce probability of wrongly rejecting null = increases probability of failing to reject null
struggles to detect stationarity when test is stationary
near root problem
DF struggles to distinguish between theta=1 and theta=0.98
structural break
sudden change in behaviour of time series eg exchange rate after BREXIT
if break occurs, DF may misinterpret as unit root
spurious regression
page 11