Transcript Regression with Time Series Data

Regression with Time Series
Data
Judge et al Chapter 15 and 16
Distributed Lag
yt  f ( xt , xt 1 , xt 2 ,
, xt n )
yt    0 xt  1 xt 1  2 xt 2 
 n xt n  et , t  n  1,
,T
Polynomial distributed lag
E ( yt )
 i   0  1i   2i 2 , i  0,
xt i
,n
0   0
1   0  1   2
 2   0  2 1  4  2
3   0  31  9 2
4   0  41  16 2
Estimating a polynomial distributed
lag
yt    0 xt  1 xt 1  2 xt 2  3 xt 3  4 xt 4  et , t  5,
0   0
1   0  1   2
i   0  1i   2i 2 , i  0,
,n
 2   0  2 1  4  2
3   0  31  9 2
4   0  41  16 2
yt     0 xt  (  0  1   2 ) xt 1  ( 0  21  4 2 ) xt 2
(  0  31  9 2 ) xt 3  (  0  41  16 2 ) xt 4  et
    0 zt 0  1 zt1   2 zt 2  et
zt 0  xt  xt 1  xt 2  xt 3  xt 4
zt1  xt 1  2 xt 2  3xt 3  4 xt 4
zt 2  xt 1  4 xt 2  9 xt 3  16 xt 4
,T
Geometric Lag

yt     i xt i  et
i 0
i  i , |  | 1
yt    0 xt  1 xt 1  2 xt 2  3 xt 3 
 et
   ( xt  xt 1  2 xt 2  3 xt 3 
)  et
Impact Multiplier: change in yt when
xt changes by one unit:

If change in xt is sustained for another
period:
  
Long-run multiplier:
(1    2  3 
)

1 
The Koyck Transformation
yt  yt 1  [  ( xt  xt 1  2 xt  2  3 xt 3 
  [  ( xt 1  xt  2  2 xt 3  3 xt  4 
 (1  )   xt  (et  et 1 )
yt  (1  )  yt 1  xt  (et  et 1 )
 1  2 yt 1  3 xt  vt
1  (1  ), 2  , 3  
vt  (et  et 1 )
)  et ]
)  et 1 ]
Autoregressive distributed lag
ARDL(1,1)
yt    0 xt  1 xt 1  1 yt 1  et
Represents an infinite distributed
lag with weights:

yt      i xt 1  ut
i 0
 0  0
1   1  10 
 2  1  1  10   11
3  121
 s  1
s 1
1
ARDL(p,q)
p
q
i 1
i 1
yt    0 xt   i xt i    i yt i  et
Approximates an infinite lag of any
shape when p and q are large.
Stationarity
• The usual properties of the least
squares estimator in a
regression using time series
data depend on the assumption
that the variables involved are
stationary stochastic processes.
• A series is stationary if its mean
and variance are constant over
time, and the covariance
between two values depends
E  yt   
only on the length of time
separating the two values
var  y   2
t
cov  yt , yt  s   cov  yt , yt s    s
Stationary Processes
yt  0.5  0.5 yt 1  N (0,1)
yt  0.5  0.9 yt 1  N (0,1)
Non-stationary processes
yt  yt 1  0.5N (0,1)
yt  yt 1  N (0,1)
Non-stationary processes with drift
yt  0.1  yt 1  0.5N (0,1)
yt  0.1  yt 1  0.5N (0,1)
Summary of time series processes
• AR(1)
yt    yt 1  vt
• Random walk
t
yt  yt 1  vt  yt   vi
i 1
• Random walk with drift
t
yt    yt 1  vt  yt  t   vi
• Deterministic trend
yt  t  vt
i 1
Trends
• Stochastic trend
–
–
–
–
Random walk
Series has a unit root
Series is integrated I(1)
Can be made stationary only by first differencing
• Deterministic trend
– Series can be made stationary either by first
differencing or by subtracting a deterministic trend.
Real data
Spurious correlation
Spurious regression
Variable
DF
B Value
Std Error
T ratio
Approx prob
Intercept
1
14.204040
0.5429
26.162
0.0001
RW2
1
-0.526263
0.00963
-54.667
0.0001
R2 0.7495
D-W 0.0305
Checking/testing for stationarity
• Correlogram
– Shows partial correlation
observations at increasing
intervals.
– If stationary these die
away.
• Box-Pierce
• Ljung-Box
• Unit root tests
Table 16.2 Correlogram for s2
Autocorrelation s AC Q-Stat
1 0.900 813.42
.|*******|
2 0.803 1461.0
.|****** |
3 0.718 1979.1
.|****** |
4 0.629 2377.9
.|***** |
5 0.545 2677.4
.|**** |
6 0.470 2900.7
.|**** |
7 0.408 3068.7
.|*** |
8 0.348 3191.2
.|*** |
9 0.299 3281.8
.|** |
10 0.266 3353.2
.|** |
Prob
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Table 16.3 Correlogram for rw1
Autocorrelation s AC Q-Stat
1 0.997 997.31
.|********
2 0.993 1988.8
.|********
3 0.990 2973.9
.|********
4 0.986 3953.2
.|********
5 0.983 4926.3
.|********
6 0.979 5893.4
.|********
7 0.975 6854.4
.|********
8 0.972 7809.4
.|*******|
9 0.968 8758.3
.|*******|
10 0.965 9701.0
.|*******|
Prob
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Unit root test
yt  yt 1  vt
var  yt   tv2
yt  yt 1  yt 1  yt 1  vt
yt     1 yt 1  vt
 yt 1  vt
H0 :   1  H0 :   0
H1 :   1  H1 :   0
Dickey Fuller Tests
• Allow for a number of possible models
– Drift
– Deterministic trend
• Account for serial correlation
Drift
yt   0  yt 1  vt
Drift against deterministic trend
yt   0  1t  yt 1  vt
Adjusting for serial correlation (ADF)
m
yt   0  yt 1   ai yt i vt
i 1
Critical values
Table 16.4 Critical Values for the Dickey-Fuller Test
Model
1%
5%
10%
yt  yt 1  vt
2.56
1.94
1.62
yt  0  yt 1  vt
3.43
2.86
2.57
yt  0  1t  yt 1  vt
3.96
3.41
3.13
2.33
1.65
1.28
Standard critical values
Example of a Dickey Fuller Test
ˆ  1.5144  .0030 PCE
PCE
t
t 1
( tau)
(-0.349) (2.557)
ˆ  2.0239  0.0152t  0.0013PCE
PCE
t
t 1
( tau)
 2 PCEt   0.9969PCEt 1
( tau)
(  18.668)
(0.1068) (0.1917) (0.1377)
ˆ  2.111  0.00397 PCE  0.2503PCE  0.0412PCE
PCE
t
t 1
t 1
t 2
( tau)
(  0.4951) (3.3068)
(  4.6594)
(  0.7679)
Cointegration
• In general non-stationary variables should
not be used in regression.
• In general a linear combination of I(1)
series, eg: et  yt  1  2 xt is I(1).
• If et is I(0) xt and yt are cointegrated and the
regression is not spurious
• et can be interpreted as the error in a longrun equilibrium.
Example of a cointegration test
Model
1%
5%
eˆt   0  eˆt 1  vt
3.90
3.34
ˆ  390.7848+1.0160DPI
PCE
t
t
(t-stats) (-24.50)
(252.97)
eˆt  0.188250  0.120344eˆt 1
(tau) (0.1107) (  4.5642)
10%
3.04