Econometrics lecture notes

1. Advanced econometrics and Stata
L9-10 Time series
Dr. Chunxia Jiang
Business School, University of Aberdeen, UK
Beijing , 17-26 Nov 2019

2. Schedule
10月17日 Evening —
L1-2 Introduction to Econometrics and Stata
10月18日 Evening —
L3-4 Data, single regression
Morning —
L5-6 (1) Hypothesis testing, Multi-regression ,
Afternoon L5-6 (2) Violation of assumptions
Morning —
L7-8 Panel data models & Endogeneity
Evening Exercises and practice
Morning —
L9-10 Time series models
Afternoon L11-12 Frontier1 SFA & practice
10月24 Evening L13-14： Frontier2 DEA & practice
10月25日 Evening L15-16 DID & practice
Morning Revision
Afternoon Exam
10月20日
10月19日
10月22日
10月26日

3. Review: panel data models
Simplest case Adding the
impact of
unknown
factors that vary
over time
Allowing
different effects
at different
points in time
Accounting for
the impact of
unknown
individual
characteristics
Accounting for
the impact of
unknown
individual
characteristics
Pooled OLS Pooled OLS Pooled OLS Least Squares
Dummy
Variables
Fixed Effect
Estimator/
Random Effect
Estimator/First
Difference
Time dummies Time dummies Time dummies Time dummies
Interaction
between
variables and
time dummies
Individual (cross-
sectional)
dummies
3

4.  Serial correlation (also called autocorrelation)
 The nature of autocorrelation
 The theoretical and practical consequences of autocorrelation
 How to detect if there is autocorrelation since et is
unobservable
 How to remedy the problem of autocorrelation
 Stationary and non-stationary
 DF test
Time series econometrics

5.  There is no correlation between two error terms (no
autocorrelation).
 OLS assumes no serial correlation
 ui and uj are independent for all i  j . i and j are two
observations over time or two cross sectional
observations
 With time series data this assumptions is very
often violated, and then we have
autocorrelation.
Assumptions of the CLRM
0
)
,
,
( 
j
i
j
i X
X
u
u
E
0
)
,
( 
j
i u
u
E

6. Autocorrelation and serial correlation
 Some authors prefer to distinguish two
 Autocorrelation: lag correlation of a given series with
itself, lagged by a number of time units
 Serial correlation: lag correlation between two different
series

7. Time series data –examples

8. Residuals
 When use this type of data in our analysis,
residuals are likely to be serially correlated
 If we plot the residuals over time we will
observe a clear pattern
 If we do a cross plot of the residuals at time t
and the residuals at time t-1 (lagged) we can
find out clearly whether there is relationship
between the two series

9. What do we mean by lagged residual?
Year residuals Reisduals(t-1)
1950 0.70
1951 0.80 0.70
1952 0.52 0.80
1953 0.43 0.52
1954 0.54 0.43
1955 -0.36 0.54
1956 0.24 -0.36
1957 0.33 0.24
1958 0.43 0.33
1959 0.33 0.43
1960 0.43 0.33
1961 0.33 0.43
1962 0.53 0.33
1963 0.23 0.53
1964 0.23 0.23
1965 0.33 0.23

10. Positive Autocorrelation
Positive Autocorrelation is indicated by a
cyclical residual plot over time.
+
-
-
t
û
+
1
ˆ 
t
u
+
-
Time
t
û
Ut-1
t
û

11. Negative Autocorrelation
Negative autocorrelation is indicated by an alternating pattern where
the residuals cross the time axis more frequently than if they were
distributed randomly
+
-
-
t
û
+
1
ˆ 
t
u
+
-
t
û
Time
Ut-1

12. No pattern in residuals – No autocorrelation
No pattern in residuals at all: this is what
we would like to see
+
t
û
-
-
+
1
ˆ 
t
u
+
-
t
û
Time
Ut-1

13. Example: the demand for ice cream

14. Model for the demand for ice cream
 Let’s assume that we try to explain the
demand for ice cream using price and income
as our explanatory variables:
 IceCreamt = α + β1pricet + β2incomet + ut
 If we now plot the actual and fitted values we
obtain the following picture:

15. Actual and fitted values for ice cream
consumption

16. Residual scatter plot

17. Serial correlation
iid
u
u
u
x
y
t
t
t
t
t
t
t












1
1
1
1 

 
We estimate this model
But the errors are not independent,
Rho is the coefficient of autocovariance
White noise errors: their expected value = 0
they have constant variance
they are not serially correlated
t
t
t u
u 
 
 1
First order autoregressive
Process or AR(1) error model
Rho can be
interpreted as the
first order
coefficient of
autocorrelation

18. Rho
 Rho can be interpreted as the first order coefficient
of autocorrelation
 Rho is a constant between -1 and +1, under AR(1)
scheme, the variance of u is still homoscedastic
 The absolute value of rho is less than 1 the AR(1)
scheme above is stationary  the mean, variance,
and covariance of u do not change over time
 if equals 1, the variance and covariance of residual are
not defined
t
t
t u
u 
 
 1 1
1 

 

19. How serial correlation affects OLS
estimation
 Similar to heteroscedasticity problem, in the
presence of serial correlation, OLS coefficient
estimates are unbiased, but inefficient (they do not
have minimum variance compared to procedures
that take into account autocorrelation).
 Statistical inference is seriously affected:
 estimated standard errors are biased. Usually these are
biased downwards (smaller than the true standard errors),
therefore the t static is upward biased.
 F and t statistics are not reliable
 R2 is unreliable as the residual variance is likely to be biased
downwards overestimate R-square

20. How to detect the presence of serial
correlation
 Durbin Watson test for first-order autoregressive error,
based on the computation of the following statistic:
 This is also called ‘d’ statistics or DW statistics: the ratio
of the sum of squared differences in successive
residuals to the RSS






 T
t
t
T
t
t
t
u
u
u
DW
1
2
2
2
1
ˆ
)
ˆ
ˆ
(
)
ˆ
1
(
2 


RSS

21. Durbin Watson test
 Easy to compute, but there are some important
assumptions underlying the test:
 The regression model has an intercept term
 Xs are fixed in repeated sampling.
 The regression does not contain lagged values of the
dependent variable
 The residuals are characterised by first order serial
correlation AR(1). If not, cannot use DW
 The error term ut is assumed to be normally distributed
 No missing observations in the data

22. Detecting Serial Correlation
 The null hypothesis:
 H0: there is no autocorrelation
 The alternative hypothesis:
 H1: there is evidence of autocorrelation
 d lies between 0 and 4 d
 d = 2 implies residuals uncorrelated
 As a rule of thumb, if d is close to 2, we do not reject the
null hypothesis we “accept” the null hypothesis
 D-W provide upper and lower bounds for d
 if d < dL then reject null of no serial correlation
 if d > dU then reject null hypothesis of no serial correlation
 if dL< d < dU then test is inconclusive
)
ˆ
1
(
2 



23. Durbin Watson test
 This test is always included in your regression output.
 Run OLS regression, Check the value for d (or DW)
 The closer d is to 0, the greater the evidence of positive serial
correlation
 The closer d is to 4, the greater the evidence of negative serial
correlation
 Find the critical dL and du from the Durbin Watson
tables for the given sample size and the given number
of explanatory variables
 Follow the decision rules as in the following table:

24. The Durbin-Watson Test: Interpreting the
Results
The inconclusive zone narrows as the sample size increases
Conditions which Must be Fulfilled for DW to be a Valid Test
1. Constant term in regression
2. Regressors are non-stochastic
3. No lags of dependent variable

25. INDEXES OF REAL WAGES AND PRODUCTIVITY IN
THE U.S. BUSINESS SECTOR, 1959-2002(1992=100)
RWAGES = Index of real compensation per hour (1992=100)
PRODUCT = Index of output per hour of all persons (1992=100)
obs RWAGES PRODUCT obs RWAGES PRODUCT
1959 59.20000 48.60000 1981 89.00000 81.90000
1960 60.70000 49.50000 1982 90.50000 81.60000
1961 62.50000 51.30000 1983 90.40000 84.50000
1962 64.60000 53.60000 1984 90.70000 86.80000
1963 66.10000 55.70000 1985 92.10000 88.50000
1964 67.70000 57.60000 1986 95.20000 91.20000
1965 69.10000 59.70000 1987 95.60000 91.60000
1966 71.70000 62.10000 1988 97.00000 93.00000
1967 73.60000 63.50000 1989 95.50000 93.90000
1968 76.00000 65.50000 1990 96.30000 95.30000
1969 77.20000 65.80000 1991 97.40000 96.40000
1970 78.60000 67.10000 1992 100.00000 100.00000
1971 80.10000 70.00000 1993 99.90000 100.50000
1972 82.30000 72.20000 1994 99.70000 101.70000
1973 84.10000 74.50000 1995 99.40000 102.30000
1974 83.10000 73.20000 1996 99.80000 105.10000
1975 83.90000 75.80000 1997 100.70000 107.40000
1976 86.20000 78.40000 1998 104.80000 110.20000
1977 87.40000 79.70000 1999 107.20000 113.00000
1978 88.90000 80.60000 2000 111.00000 116.50000
1979 89.10000 80.50000 2001 112.10000 118.80000
1980 88.90000 80.30000 2002 113.50000 125.10000

26. Example
 Regressing real wages on productivity, we obtain the following
results:
Realwages= 29.575 + 0.701Productivity
se (1.461) (0.017)
R2 = 0.976
DW = 0.214
 d=0.214,
 No. of observations: 44
 No. of explanatory variables: 1
 From the Durbin Watson tables (5% significance level): dL=1.475,
du=1.566
 d<dL :There is evidence of positive serial correlation.

27. What can we do about first order
serial correlation?
 Four options:
 Find out if the autocorrelation is pure autocorrelation
and not the result of mis-specification of the model, i.e.
excluding some important variables.
 If it is pure autocorrelation  appropriate
transformation. GLS method
 In large samples, use Newey-West method to obtain
standard errors of OLS estimator that are corrected for
autocorrelation
 Some situations, we can use the OLS method

28. What if the model contains the lagged
dependent variable?
t
t
t
t u
x
y
y 


  2
1
1 


Savings Lagged savings Income
The d statistics in these models is biased towards 2 so there is a bias against
discovering first order serial correlation
The inclusion of lagged dependent variables causes a problem of higher
Order Serial correlation.

29. How to detect the presence of serial
correlation
 The Breusch-Godfrey LM (Lagrange Multiplier)
test of higher-order autocorrelation
 Suppose the disturbance term et is generated by
the following pth-order autoregressive process,
AR(p):
where vt is purely random disturbance term with
zero mean and constant variance.
(7)
v
e
ρ
.
.
.
e
ρ
e
ρ
e t
p
t
p
2
t
2
1
t
1
t 



 



30. Breusch and Godfrey test
 The null hypothesis: all the coefficients in (7) are equal to zero.
 The alternative hypothesis: at least one of the coefficients is not
equal to zero.
 The null hypothesis can be tested as follows:
 Estimate the original regression (OLS) to obtain .
 Regress on all the regressors (X’s) and the terms in (7). For
example, if the original model has only one X variable, we run the
following auxiliary regression to obtain R2:
 If the sample size is large, then
 If the calculated value is greater than the critical value of chi-square,
we reject the hypothesis that there is no serial correlation.
t
ê
t
ê
(7a)
v
e
ρ
.
.
.
e
ρ
e
ρ
X
ê t
p
t
p
2
t
2
1
t
1
2t
2
1
t 





 




2
2
)
( p
R
p
n 



31. Correcting for Higher Order Serial
Correlation
 No easy answer
 Typically indicative of a mis-specification of
original equation
 Often due to omitted lags of explained and/or
explanatory variables (i.e. misspecified dynamics)
 Try including more lags of the variables currently in the
equation or (lags of) new variables
 Can use Newey-West std errors to account for
serial correlation