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 yi i i n

x

0 0

1 0 1 2

2 0 1 2

3 0 1 2

4 0 1 2

2 4

3 9

4 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 4

3 9

4 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 x

x x e

z z z e

0 1 2 3 4

1 1 2 3 4

2 1 2 3 4

2 3 4

4 9 16

t t t t t t

t t t t t

t t t t t

z x x x x x

z x x x x

z 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 e

y x v

1 2 3(1 ), ,

1( )t t tv e e

Autoregressive distributed lag

ARDL(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 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 0.7495D-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 H

H 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