Consider a simulation based on two i.i.d. time series Y0;t and X0;t that have joint distribution
Y0;t
X0;t
N
0
0
;
1
1
:
The value for may be either = 0 or = 0:5. In the simulation, 10,000 repeated samples of
size n are drawn, and for each of these an OLS regression of Y0;t on an intercept and X0;t is
estimated, followed by the usual t-test of the signi cance of the slope coe cient on X0;t. The
rejection frequencies of these tests are given in the left half of the table below (headed Y0;t;X0;t).
In addition, two time series Y1;t and X1;t are de ned by
Y1;t = Y1;t?1 + Y0;t
X1;t = X1;t?1 + X0;t
with starting values Y1;t = X1;t = 0 for t = 0. Similarly to the Y0;t;X0;t samples, an OLS
regression of Y1;t on an intercept and X1;t is estimated, followed by the usual t-test of the
signi cance of the slope coe cient on X1;t. The simulated rejection frequencies of these tests
are given in the right half of the table below (headed Y1;t;X1;t).
Simulated rejection frequencies of the regression t-tests
Y0;t;X0;t Y1;t;X1;t
n = 0 = 0:5 = 0 = 0:5
50 0.049 0.959 0.669 0.782
100 0.051 0.991 0.759 0.850
200 0.050 1.000 0.839 0.896
1. Are the Y0;t and X0;t time series
(a) (2 marks) autocorrelated?
(b) (3 marks) stationary?
(c) (2 marks) I(1)?
(d) (2 marks) cointegrated with each other?
Brie
y justify each answer.
1
2. Are the Y1;t and X1;t time series
(a) (2 marks) autocorrelated?
(b) (2 marks) stationary?
(c) (2 marks) I(1)?
(d) (4 marks) cointegrated with each other?
Brie
y justify each answer.
3. We use here the second concept of a spurious regression” given in week 9 lectures |
the nding of a signi cant statistical relationship between two variables when in fact no
such relationship exists in the population. (The presence or otherwise of causality does
not enter the current discussion.) Applying this concept of spurious regression” to the
current simulation setting:
(a) (2 marks) in which situation(s) in the simulation would a regression between Y0;t
and X0;t be deemed spurious?
(b) (2 marks) in which situation(s) in the simulation would a regression between Y0;t
and X0;t be deemed genuine”1?
(c) (2 marks) in which situation(s) in the simulation would a regression between Y1;t
and X1;t be deemed spurious?
(d) (2 marks) in which situation(s) in the simulation would a regression between Y1;t
and X1;t be deemed genuine”?
4. Consider the simulation results in the table above.
(a) (1 mark) For each n = 50; 100; 200, what are the simulated probabilities of nding a
spurious relationship between Y0;t and X0;t?
(b) (1 mark) For each n = 50; 100; 200, what are the simulated probabilities of nding a
spurious relationship between Y1;t and X1;t?
(c) (1 mark) For each n = 50; 100; 200, what are the simulated probabilities of nding a
genuine relationship between Y0;t and X0;t?
(d) (1 mark) For each n = 50; 100; 200, what are the simulated probabilities of nding a
genuine relationship between Y1;t and X1;t?
5. (a) (4 marks) Based on the simulation results, why is there such concern about the
danger of spurious regressions involving I(1) time series, but not involving I(0) time
series?
(b) (5 marks) Suppose you wanted to test for a linear relationship between Y1;t and X1;t.
What would be a better way than simply regressing Y1;t on an intercept and X1;t?
Using on the simulation results, explain why you suggestion would be better.
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