Transcript Tests of Nonstationarity: Other tests

Type
author name/s here
Dougherty
Introduction to Econometrics,
5th edition
Chapter heading
Chapter 13: Introduction to
Nonstationary Time Series
© Christopher Dougherty, 2016. All rights reserved.
TESTS OF NONSTATIONARITY: OTHER TESTS
General model
Yt   1   2Yt 1   t   t
Case (a)
Yt   1   2Yt 1   t
Case (b)
Yt  Yt 1   t
Dickey and Fuller proposed two further tests for nonstationarity, one involving a scaled
estimator of the slope coefficient, T ˆ2   2 , and the other an F test.


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TESTS OF NONSTATIONARITY: OTHER TESTS
General model
Yt   1   2Yt 1   t   t
Case (a)
Yt   1   2Yt 1   t
Case (b)
Yt  Yt 1   t
We will confine the discussion to the case where the process for Yt has only one lag. Both
tests can be generalized in the same way as the augmented Dickey–Fuller t test in the
previous section.
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TESTS OF NONSTATIONARITY: OTHER TESTS
General model
Yt   1   2Yt 1   t   t
Case (a)
Yt   1   2Yt 1   t
Case (b)
Yt  Yt 1   t


We will begin with the test using the scaled estimator T ˆ2   2 and initially we will
suppose that there is no evidence of a trend in the data. Our task is to discriminate between
Case (a) and Case (b).
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TESTS OF NONSTATIONARITY: OTHER TESTS
General model
Yt   1   2Yt 1   t   t
Case (a)
Yt   1   2Yt 1   t
Case (b)
Yt  Yt 1   t
Case (b) is a special case of
Yt   2Yt 1   t
The Dickey–Fuller test using the scaled estimator exploits the fact that 2 in the equation
Yt   2Yt 1   t is a pure number. Yt‒1 must have the same dimensions as Yt, and so 2 is
dimensionless.
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TESTS OF NONSTATIONARITY: OTHER TESTS
General model
Yt   1   2Yt 1   t   t
Case (a)
Yt   1   2Yt 1   t
Case (b)
Yt  Yt 1   t
Case (b) is a special case of
Yt   2Yt 1   t
It follows that if we fit the model using OLS, ̂ 2 logically must also be dimensionless.
(Verification is left as Exercise 13.16.)
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TESTS OF NONSTATIONARITY: OTHER TESTS
General model
Yt   1   2Yt 1   t   t
Case (a)
Yt   1   2Yt 1   t
Case (b)
Yt  Yt 1   t
Case (b) is a special case of
Yt   2Yt 1   t
The shape of the distribution of ̂ 2 depends only on the sample size, and this means that we
can use the distribution to make a direct test of hypotheses relating to 2.
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TESTS OF NONSTATIONARITY: OTHER TESTS
General model
Yt   1   2Yt 1   t   t
Case (a)
Yt   1   2Yt 1   t
Case (b)
Yt  Yt 1   t
Case (b) is a special case of
Yt   2Yt 1   t
Of course, the critical value of the test depends on the size of the sample since this
determines the variance of the distribution.
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TESTS OF NONSTATIONARITY: OTHER TESTS
General model
Yt   1   2Yt 1   t   t
Case (a)
Yt   1   2Yt 1   t
Case (b)
Yt  Yt 1   t
Case (b) is a special case of
Yt   2Yt 1   t
Therefore it is more sensible to perform the test, not directly on ̂ 2, but on a scaled version
that has a limiting distribution. The critical values will then be less sensitive to the sample
size and will converge to a limit.
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TESTS OF NONSTATIONARITY: OTHER TESTS
Yt   1   2Yt 1   t   t
General model
Case (a)
Yt   1   2Yt 1   t
Case (b)
Yt  Yt 1   t
Case (b) is a special case of
Yt   2Yt 1   t


d
T ˆ2   2 
 N  0, 1   22 
For this we return to the theory in Section 11.5, where we investigated the properties of the
OLS estimator of ̂ 2. We saw that the estimator is consistent and that, provided that | 2 | < 1,
its transformation T ˆ2   2 has the limiting normal distribution shown.


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TESTS OF NONSTATIONARITY: OTHER TESTS
General model
Yt   1   2Yt 1   t   t
Case (a)
Yt   1   2Yt 1   t
Case (b)
Yt  Yt 1   t
Case (b) is a special case of
Yt   2Yt 1   t


d
T ˆ2   2 
 N  0, 1   22 
What happens if 2 = 1, and the process is a random walk? If we try to use the theory when 2
= 1, we immediately get into trouble. It implies that the limiting distribution of the scaled
estimator has zero variance.
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TESTS OF NONSTATIONARITY: OTHER TESTS
Random walk: distribution of ˆ 2
Yt  Yt 1   t
Yˆt  ˆ1  ˆ2 X t
30
T = 200
20
T = 100
10
T = 50
T = 25
0
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Distribution of ˆ 2
This is actually correct. To understand what is happening, we will look at the distribution of
the estimated values of 2, when 1 = 0 and 2 = 1, for T = 25, 50, 100, and 200. The figure
shows the distribution of ˆ 2 for the different sample sizes when we fit Yt   1   2Yt 1   t .
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TESTS OF NONSTATIONARITY: OTHER TESTS
Random walk: distribution of ˆ 2
Yt  Yt 1   t
Yˆt  ˆ1  ˆ2 X t
30
T = 200
20
T = 100
10
T = 50
T = 25
0
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Distribution of ˆ 2
As expected, the OLS estimator is biased for finite samples but consistent. The distribution
collapses to a spike as the sample size increases, and the spike is at the true value 2 = 1.
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TESTS OF NONSTATIONARITY: OTHER TESTS
Random walk: distribution of ˆ 2
Yt  Yt 1   t
Yˆt  ˆ1  ˆ2 X t
30
T = 200
20
T = 100
10
T = 50
T = 25
0
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Distribution of ˆ 2
This is as anticipated since OLS is a consistent estimator for all AR(1) processes. What is
surprising is the rate at which the distribution contracts. In all applications so far, the
variance of the OLS estimator has been inversely proportional to the size of the sample.
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TESTS OF NONSTATIONARITY: OTHER TESTS
Random walk: distribution of ˆ 2
Yt  Yt 1   t
Yˆt  ˆ1  ˆ2 X t
30
T = 200
20
T = 100
10
T = 50
T = 25
0
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Distribution of ˆ 2
Hence the standard deviation is inversely proportional to √T, and since the area is constant
at 1, the height of the distribution is proportional to √T.
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TESTS OF NONSTATIONARITY: OTHER TESTS
For comparison: distribution of ˆ 2 when Yt and Xt are independent white noise
~ N 0,1
7
Yt , X t
Yˆt  ˆ1  ˆ2 X t
6
T = 200
5
T = 100
ˆ 2 is √T consistent
4
T = 50
3
2
T = 25
1
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Distribution of ˆ 2
We saw a typical example in the slideshow on spurious regressions. This figure shows the
distribution of ˆ 2 for the regression where Yt and Xt were iid white noise processes drawn
independently from a normal distribution with zero mean and unit variance.
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TESTS OF NONSTATIONARITY: OTHER TESTS
For comparison: distribution of ˆ 2 when Yt and Xt are independent white noise
~ N 0,1
7
Yt , X t
Yˆt  ˆ1  ˆ2 X t
6
T = 200
5
T = 100
ˆ 2 is √T consistent
4
T = 50
3
2
T = 25
1
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Distribution of ˆ 2
You can see that, as the sample size doubles, the height is increased by a factor √2 = 1.41
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TESTS OF NONSTATIONARITY: OTHER TESTS
Random walk: distribution of ˆ 2
Yt  Yt 1   t
Yˆt  ˆ1  ˆ2 X t
30
T = 200
20
ˆ 2 is T consistent
T = 100
10
T = 50
T = 25
0
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Distribution of ˆ 2
However, when 2 = 1 and the true process is a random walk, it can be seen that the height
is proportional to T. The height doubles from T = 25 to T = 50, it doubles again from T = 50
to T = 100, and it doubles once more from T = 100 to T = 200.
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TESTS OF NONSTATIONARITY: OTHER TESTS
Random walk: distribution of ˆ 2
Yt  Yt 1   t
Yˆt  ˆ1  ˆ2 X t
30
T = 200
20
ˆ 2 is T consistent
T = 100
10
T = 50
T = 25
0
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Distribution of ˆ 2
This means that the variance of the distribution of ˆ 2 is inversely proportional to T 2, not T.
Multiplying by √T is not enough to prevent the distribution contracting.
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TESTS OF NONSTATIONARITY: OTHER TESTS

Random walk: distribution of T ˆ2  1



d
T ˆ2   2 
 N  0, 1   22 
Yt  Yt 1   t
Yˆt  ˆ1  ˆ2 X t
2
T = 200
T = 100
1
T = 50
T = 25
0
-5
-4
-3
-2
-1
0

Distribution of T ˆ2  1

1
2


This figure shows the distribution of T ˆ2  1 . Although we have scaled by a factor √T,
the distribution still collapses to a spike instead of a limiting normal distribution.
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TESTS OF NONSTATIONARITY: OTHER TESTS
Random walk: distribution of T  ˆ2  1


d
T ˆ2   2 
 N  0, 1   22 
Yt  Yt 1   t
Yˆt  ˆ1  ˆ2 X t
2
T = 200
T = 100
1
T = 50
T = 25
0
-5
-4
-3
-2
-1
0

Distribution of T ˆ2  1
1
2

Because the distribution is contracting to the true value faster than the standard rate, the
estimator is described as superconsistent. When 2 = 1, to obtain a limiting distribution
related to the OLS estimator, we must consider T ˆ2  1 , not T ˆ2  1 .




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TESTS OF NONSTATIONARITY: OTHER TESTS
Random walk: distribution of T  ˆ2  1
Yt  Yt 1   t
Yˆt  ˆ1  ˆ2 X t
T = 25
0.1
T = 200
0
-30
-25
-20
-15
-10
-5

Distribution of T ˆ2  1

0
5


This figure shows the distribution of T ˆ2  1 . The statistic does have a limiting
distribution, and reaches it quite quickly, since the distribution for T = 25 is almost the same
as that for T = 200. (The distributions for T = 50 and T = 100 have been omitted, for clarity.)
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TESTS OF NONSTATIONARITY: OTHER TESTS
Random walk: distribution of T  ˆ2  1
Yt  Yt 1   t
Yˆt  ˆ1  ˆ2 X t
T = 25
0.1
T = 200
0
-30
-25
-20
-15
-10
-5

Distribution of T ˆ2  1
0
5

However, we now have a further surprise. The distribution is not normal. This does not
give rise to any practical problem. The critical values for the limiting distribution, as for
those for finite samples, are easily established using simulation methods.
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TESTS OF NONSTATIONARITY: OTHER TESTS
Direct test using the distribution of T  ˆ2  1
Yt  Yt 1   t
Yˆt  ˆ1  ˆ2 X t
T = 25
0.1
T = 200
0
-30
-25
-20
-15
-10
-5

Distribution of T ˆ2  1
0
5

Since we can rule out 2 > 1, we can perform a one-sided test with the null and alternative
hypotheses being H0: 2 = 1 and H1: 2 < 1.
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TESTS OF NONSTATIONARITY: OTHER TESTS
Direct test using the distribution of T  ˆ2  1
Yt  Yt 1   t
Yˆt  ˆ1  ˆ2 X t
T→∞
5% limit = –14.1
1% limit = –20.6
T = 25
0.1
T = 200
0
-30
-25
-20
-15
-10
-5

Distribution of T ˆ2  1
0
5

For the limiting distribution, the critical values are –14.1 and –20.6 at the 5 percent and 1
percent levels. Finite-sample distributions, established through simulation, have thinner
left tails and so the critical values are smaller.
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TESTS OF NONSTATIONARITY: OTHER TESTS
Direct test using the distribution of T  ˆ2  1
Yt  Yt 1   t
Yˆt  ˆ1  ˆ2 X t
T = 25
5% limit = –12.1
1% limit = –16.6
0.1
limit of 5% rejection region
limit of 1% rejection region
0.0
-30
-25
-20
-15
-10
-5

Distribution of T ˆ2  1
0
5

This figure shows the rejection regions for one-sided 5 percent (limit –12.1) and 1 percent
(limit –16.6) tests for T = 25. Table A.7 gives the critical values of T ˆ2  1 for finite
samples.


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TESTS OF NONSTATIONARITY: OTHER TESTS
F test, untrended data
H0: random walk
Yt  Yt 1   t
H1: stationary process
Yt   1   2Yt 1   t
2  1
H 0:  1  0 and  2  1
We next come to the Dickey–Fuller F test. If there is no evidence of a trend in the graph of a
process, we need to consider whether the process is better characterized as a stationary
process Yt   1   2Yt 1   t with | 2 | < 1, or a random walk Yt  Yt 1   t .
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TESTS OF NONSTATIONARITY: OTHER TESTS
F test, untrended data
H0: random walk
Yt  Yt 1   t
H1: stationary process
Yt   1   2Yt 1   t
2  1
H 0:  1  0 and  2  1
There is an asymmetry in the specification under the two possibilities. If the process is a
random walk, there must be no intercept. Otherwise the process would be a random walk
with drift and therefore trended.
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TESTS OF NONSTATIONARITY: OTHER TESTS
F test, untrended data
H0: random walk
Yt  Yt 1   t
H1: stationary process
Yt   1   2Yt 1   t
2  1
H 0:  1  0 and  2  1
If the process is stationary, the intercept may, and in general, will, be nonzero. For this
reason, in the discussion below, it should be assumed that the fitted model includes an
intercept and that we are interested in testing the restrictions 1 = 0 and 2 = 1.
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TESTS OF NONSTATIONARITY: OTHER TESTS
F test, untrended data
H0: random walk
Yt  Yt 1   t
H1: stationary process
Yt   1   2Yt 1   t
2  1
H 0:  1  0 and  2  1
The Dickey–Fuller F test exploits the fact that there are two restrictions. To test the joint
restrictions, it is sufficient to construct the OLS F statistic in the usual way.
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TESTS OF NONSTATIONARITY: OTHER TESTS
F test, untrended data
H0: random walk
Yt  Yt 1   t
H1: stationary process
Yt   1   2Yt 1   t
2  1
H 0:  1  0 and  2  1
However, as might be anticipated, the F statistic does not have a standard distribution.
Table A.8 provides critical values.
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TESTS OF NONSTATIONARITY: OTHER TESTS
F test, trended data
H0: random walk with drift
Yt   1  Yt 1   t
H1: deterministic trend
Yt   1   2Yt 1   t   t
2  1
H 0:  2  1 and   0
If an inspection of the graph of a process reveals evidence of a trend, we need to consider
whether the process is better characterized as a random walk with drift Yt   1  Yt 1   t
with 1 ≠ 0, or a deterministic trend Yt   1   2Yt 1   t   t with | 2 | < 1 .
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TESTS OF NONSTATIONARITY: OTHER TESTS
F test, trended data
H0: random walk with drift
Yt   1  Yt 1   t
H1: deterministic trend
Yt   1   2Yt 1   t   t
2  1
H 0:  2  1 and   0
1 is now unconstrained, but we again have two restrictions that could be tested with an F
test. We have argued that a process cannot combine a random walk with drift and a time
trend, so we can test the composite hypothesis H0 : 2 = 1,  = 0.
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TESTS OF NONSTATIONARITY: OTHER TESTS
F test, trended data
H0: random walk with drift
Yt   1  Yt 1   t
H1: deterministic trend
Yt   1   2Yt 1   t   t
2  1
H 0:  2  1 and   0
Critical values for this F test are also given in Table A.8 at the end of the text.
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TESTS OF NONSTATIONARITY: OTHER TESTS
Power of the tests
Probability of rejecting H0: 2 = 1
1.0
0.8
scaled ^
2
F
0.6
t
0.4
0.2
size of test: 5%
0.0
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Value of 2
We have now considered three different Dickey–Fuller tests for nonstationarity and there is
no guarantee that they will lead to the same conclusion. There are two obvious questions
that we should ask. How good are these tests, and which is the best?
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TESTS OF NONSTATIONARITY: OTHER TESTS
Power of the tests
Probability of rejecting H0: 2 = 1
1.0
0.8
scaled ^
2
F
0.6
t
0.4
0.2
size of test: 5%
0.0
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Value of 2
To answer these questions, we need to be more specific about what we mean by ‘good’ and
‘best’. For any given significance level, we would like a test to have the least risk of a Type
II error, that is, of failing to reject the null hypothesis when it is false.
35
TESTS OF NONSTATIONARITY: OTHER TESTS
Power of the tests
Probability of rejecting H0: 2 = 1
1.0
0.8
scaled ^
2
F
0.6
t
0.4
0.2
size of test: 5%
0.0
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Value of 2
The power of a test is defined to be probability of rejecting the null hypothesis when it is
false, so we want our tests to have high power, and the best test will be that with the highest
power.
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TESTS OF NONSTATIONARITY: OTHER TESTS
Power of the tests
Probability of rejecting H0: 2 = 1
1.0
0.8
scaled ^
2
F
0.6
t
0.4
0.2
size of test: 5%
0.0
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Value of 2
We will consider this issue for the case where the process is a stationary autoregressive
process. What is the probability of rejecting the (false) null hypothesis that the process is a
random walk? We fit the model Yt   1   2Yt 1   t for | 2 | < 1 and test H0: 2 = 1.
37
TESTS OF NONSTATIONARITY: OTHER TESTS
Power of the tests
Probability of rejecting H0: 2 = 1
1.0
0.8
scaled ^
2
F
0.6
t
0.4
0.2
size of test: 5%
0.0
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Value of 2
The figure shows the power of the three tests as a function of 2, using a 5 percent
significance level for each of them, with T = 100.
38
TESTS OF NONSTATIONARITY: OTHER TESTS
Power of the tests
Probability of rejecting H0: 2 = 1
1.0
0.8
scaled ^
2
F
0.6
t
0.4
0.2
size of test: 5%
0.0
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Value of 2


The figure seems to answer our questions. The test based on T ˆ2  1 has greatest power
for all values of 2 < 1. The t test is next in terms of power, and the F is test least powerful.
39
TESTS OF NONSTATIONARITY: OTHER TESTS
Power of the tests
Probability of rejecting H0: 2 = 1
1.0
0.8
scaled ^
2
F
0.6
t
0.4
0.2
size of test: 5%
0.0
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Value of 2
The tests all have high power when 2 < 0.8. For values above 0.8, the power rapidly
diminishes and becomes low.
40
TESTS OF NONSTATIONARITY: OTHER TESTS
Power of the tests
Probability of rejecting H0: 2 = 1
1.0
0.8
scaled ^
2
F
0.6
t
0.4
0.2
size of test: 5%
0.0
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Value of 2


For 2 = 0.9, for example, even the test based on T ˆ2  1 will fail to reject the null
hypothesis 50 percent of the time. This demonstrates the difficulty of discriminating
between nonstationary processes and stationary processes that are highly autoregressive.
41
TESTS OF NONSTATIONARITY: OTHER TESTS
Power of the tests
Probability of rejecting H0: 2 = 1
1.0
0.8
scaled ^
2
F
0.6
t
0.4
0.2
size of test: 5%
0.0
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Value of 2
The inferiority of the F test in this comparison may come as a surprise, given that it is
making more use of the data than the other two.
42
TESTS OF NONSTATIONARITY: OTHER TESTS
Power of the tests
Probability of rejecting H0: 2 = 1
1.0
0.8
scaled ^
2
F
0.6
t
0.4
0.2
size of test: 5%
0.0
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Value of 2
However, there is a simple reason for this. The other two tests are one-sided and this
increases their power, holding the significance level constant.
43
TESTS OF NONSTATIONARITY: OTHER TESTS
Test of a deterministic trend
Distribution of ˆ
Distribution of ˆ
T = 100
100
100
50
50
T = 100
T = 50
T = 50
T = 25
T = 25
0
0
0
Fitted model:
0.1
0.2
0.3
Yˆt  ˆ1  ˆ t
0.4
0
0.1
0.2
0.3
0.4
Yˆt  ˆ1  ˆ2Yt 1  ˆ t
We now consider tests of deterministic trends. Tests of nonstationarity attributable to the
presence of a deterministic trend are relatively straightforward. The simple model
Yt   1   t   t may be fitted using OLS.
44
TESTS OF NONSTATIONARITY: OTHER TESTS
Test of a deterministic trend
Distribution of ˆ
Distribution of ˆ
T = 100
100
100
50
50
T = 100
T = 50
T = 50
T = 25
T = 25
0
0
0
Fitted model:
0.1
0.2
0.3
Yˆt  ˆ1  ˆ t
0.4
0
0.1
0.2
0.3
0.4
Yˆt  ˆ1  ˆ2Yt 1  ˆ t
The OLS estimator of the slope coefficient is hyperconsistent, meaning that is variance is
inversely proportional to T 3, but this does not affect the diagnostic and test statistics.
These remain valid in finite samples.
45
TESTS OF NONSTATIONARITY: OTHER TESTS
Test of a deterministic trend
Distribution of ˆ
Distribution of ˆ
T = 100
100
100
50
50
T = 100
T = 50
T = 50
T = 25
T = 25
0
0
0
Fitted model:
0.1
0.2
0.3
Yˆt  ˆ1  ˆ t
0.4
0
0.1
0.2
0.3
0.4
Yˆt  ˆ1  ˆ2Yt 1  ˆ t
This is illustrated for the case  = 0.2 in the left chart in the figure. Since the standard
deviation of the distribution is inversely proportional to T3/2, the height is proportional to T3/2,
and so it more than doubles when the sample size is doubled.
46
TESTS OF NONSTATIONARITY: OTHER TESTS
Test of a deterministic trend
Distribution of ˆ
Distribution of ˆ
T = 100
100
100
50
50
T = 100
T = 50
T = 50
T = 25
T = 25
0
0
0
Fitted model:
0.1
0.2
0.3
Yˆt  ˆ1  ˆ t
0.4
0
0.1
0.2
0.3
0.4
Yˆt  ˆ1  ˆ2Yt 1  ˆ t
More often, one may wish to consider the more general model of a stationary
autoregressive process around a deterministic trend, fitting Yt   1   2Yt 1   t   t
with | 2 | < 1.
47
TESTS OF NONSTATIONARITY: OTHER TESTS
Test of a deterministic trend
Distribution of ˆ
Distribution of ˆ
T = 100
100
100
50
50
T = 100
T = 50
T = 50
T = 25
T = 25
0
0
0
Fitted model:
0.1
0.2
0.3
Yˆt  ˆ1  ˆ t
0.4
0
0.1
0.2
0.3
0.4
Yˆt  ˆ1  ˆ2Yt 1  ˆ t
The OLS estimator of then reverts to being √T consistent, as illustrated in the right chart in
the figure.
48
TESTS OF NONSTATIONARITY: OTHER TESTS
Test of a deterministic trend
Distribution of ˆ
Distribution of ˆ
T = 100
100
100
50
50
T = 100
T = 50
T = 50
T = 25
T = 25
0
0
0
Fitted model:
0.1
0.2
0.3
Yˆt  ˆ1  ˆ t
0.4
0
0.1
0.2
0.3
0.4
Yˆt  ˆ1  ˆ2Yt 1  ˆ t
As usual, given the use of the lagged dependent variable as a regressor, OLS estimators are
consistent, rather than unbiased, and test statistics are valid only asymptotically.
49
TESTS OF NONSTATIONARITY: OTHER TESTS
Test of a deterministic trend
Distribution of ˆ
Distribution of ˆ
T = 100
100
100
50
50
T = 100
T = 50
T = 50
T = 25
T = 25
0
0
0
Fitted model:
0.1
0.2
0.3
Yˆt  ˆ1  ˆ t
0.4
0
0.1
0.2
0.3
0.4
Yˆt  ˆ1  ˆ2Yt 1  ˆ t
Section 13.5 ends with some further comments on tests that do not lend themselves to a
graphical treatment and so will not be discussed here.
50
Copyright Christopher Dougherty 2016.
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Introduction to Econometrics, fifth edition 2016, Oxford University Press.
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2016.05.26