regression with time series data
DESCRIPTION
Regression with Time Series Data. Judge et al Chapter 15 and 16. Distributed Lag. Polynomial distributed lag. Estimating a polynomial distributed lag. Geometric Lag. Impact Multiplier: change in y t when x t changes by one unit:. If change in x t is sustained for another period:. - PowerPoint PPT PresentationTRANSCRIPT
Regression with Time Series Data
Judge et al Chapter 15 and 16
Distributed Lag
1 2( , , , , )t t t t t ny f x x x x
0 1 1 2 2 , 1, ,t t t t n t n ty x x x x e t n T
Polynomial distributed lag2
0 1 2( ) , 0, ,t
it i
E y i i i nx
0 0
1 0 1 2
2 0 1 2
3 0 1 2
4 0 1 2
2 43 94 16
Estimating a polynomial distributed lag
0 1 1 2 2 3 3 4 4 , 5, ,t t t t t t ty x x x x x e t T
0 0
1 0 1 2
2 0 1 2
3 0 1 2
4 0 1 2
2 43 94 16
0 0 1 2 1 0 1 2 2
0 1 2 3 0 1 2 4
0 0 1 1 2 2
( ) ( 2 4 )( 3 9 ) ( 4 16 )
t t t t
t t t
t t t t
y x x xx x e
z z z e
0 1 2 3 4
1 1 2 3 4
2 1 2 3 4
2 3 44 9 16
t t t t t t
t t t t t
t t t t t
z x x x x xz x x x xz x x x x
20 1 2 , 0, ,i i i i n
Geometric Lag
0t i t i t
i
y x e
If change in xt is sustained for another period:
, | | 1ii
Impact Multiplier: change in yt when xt changes by one unit:
0 1 1 2 2 3 3
2 31 2 3( )
t t t t t t
t t t t t
y x x x x e
x x x x e
Long-run multiplier:
2 3(1 )1
The Koyck Transformation2 3
1 1 2 3
2 31 2 3 4 1
1
[ ( ) ]
[ ( ) ]
(1 ) ( )
t t t t t t t
t t t t t
t t t
y y x x x x e
x x x x e
x e e
1 1
1 2 1 3
(1 ) ( )t t t t t
t t t
y y x e ey x v
1 2 3(1 ), ,
1( )t t tv e e
Autoregressive distributed lagARDL(1,1)
0 1 1 1 1t t t t ty x x y e
Represents an infinite distributed lag with weights:
10
t i t ti
y x u
0 0
1 1 1 0
2 1 1 1 0 1 1
23 1 1
11 1
ss
01 1
p q
t t i t i i t i ti i
y x x y e
ARDL(p,q)
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 only on the length of time separating the two values
tE y
2var ty
cov , cov ,t t s t t s sy y y y
Stationary Processes
10.5 0.5 (0,1)t ty y N
10.5 0.9 (0,1)t ty y N
Non-stationary processes
1 0.5 (0,1)t ty y N
1 (0,1)t ty y N
Non-stationary processes with drift
10.1 0.5 (0,1)t ty y N
10.1 0.5 (0,1)t ty y N
• AR(1)
• Random walk
• Random walk with drift
• Deterministic trend
Summary of time series processes
11
t
t t t t ii
y y v y t v
11
t
t t t t ii
y y v y v
1t t ty y v
t ty t v
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 regressionVariable 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 0.7495D-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 Prob .|*******| 1 0.900 813.42 0.000 .|****** | 2 0.803 1461.0 0.000 .|****** | 3 0.718 1979.1 0.000 .|***** | 4 0.629 2377.9 0.000 .|**** | 5 0.545 2677.4 0.000 .|**** | 6 0.470 2900.7 0.000 .|*** | 7 0.408 3068.7 0.000 .|*** | 8 0.348 3191.2 0.000 .|** | 9 0.299 3281.8 0.000 .|** | 10 0.266 3353.2 0.000
Table 16.3 Correlogram for rw1 Autocorrelation s AC Q-Stat Prob .|******** 1 0.997 997.31 0.000 .|******** 2 0.993 1988.8 0.000 .|******** 3 0.990 2973.9 0.000 .|******** 4 0.986 3953.2 0.000 .|******** 5 0.983 4926.3 0.000 .|******** 6 0.979 5893.4 0.000 .|******** 7 0.975 6854.4 0.000 .|*******| 8 0.972 7809.4 0.000 .|*******| 9 0.968 8758.3 0.000 .|*******| 10 0.965 9701.0 0.000
Unit root test1t t ty y v
2var t vy t
1 1 1
1
1
1t t t t t
t t t
t t
y y y y v
y y v
y v
0 0
1 1
: 1 : 0: 1 : 0
H HH H
Dickey Fuller Tests• Allow for a number of possible models
– Drift– Deterministic trend
• Account for serial correlation
0 1t t ty y v
0 1 1t t ty t y v
0 11
m
t t i t i ti
y y a y v
Drift
Drift against deterministic trend
Adjusting for serial correlation (ADF)
Table 16.4 Critical Values for the Dickey-Fuller Test
Model 1% 5% 10%
2.56 1.94 1.62
3.43 2.86 2.57
3.96 3.41 3.13
Standard critical values 2.33 1.65 1.28
Critical values
1t t ty y v
0 1t t ty y v
0 1 1t t ty t y v
Example of a Dickey Fuller Test
1ˆ 1.5144 .0030
( tau) (-0.349) (2.557)t tPCE PCE
1ˆ 2.0239 0.0152 0.0013
( tau) (0.1068) (0.1917) (0.1377)t tPCE t PCE
1 1 2ˆ 2.111 0.00397 0.2503 0.0412
( tau) ( 0.4951) (3.3068) ( 4.6594) ( 0.7679)t t t tPCE PCE PCE PCE
210.9969
( tau) ( 18.668)t tPCE PCE
Cointegration
• In general non-stationary variables should not be used in regression.
• In general a linear combination of I(1) series, eg: 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 long-run equilibrium.
1 2t t te y x
Example of a cointegration test
0 1ˆ ˆt t te e v
Model 1% 5% 10%
3.90 3.34 3.04
ˆ 390.7848+1.0160(t-stats) (-24.50) (252.97)
t tPCE DPI
1ˆ ˆ0.188250 0.120344(tau) (0.1107) ( 4.5642)
t te e