Auto Correlation Presentation

BS Statistics
6th
Semester
Regular
University of Sargodha
Session 2011-2015

 Introduction
 Causes of Autocorrelation
 OLS Estimation
 BLUE Estimator
 Consequences of using OLS
 Detecting Autocorrelation

1. Introduction
Autocorrelation occurs in time-series studies
when the errors associated with a given time
period carry over into future time periods.
For example, if we are predicting the growth of
stock dividends, an overestimate in one year is
likely to lead to overestimates in succeeding
years.

1. Introduction
Times series data follow a natural
ordering over time.
It is likely that such data exhibit
intercorrelation, especially if the time
interval between successive observations
is short, such as weeks or days.

1. Introduction
We expect stock market prices to move or
move down for several days in succession.
In situation like this, the assumption of no auto
or serial correlation in the error term that
underlies the CLRM will be violated.
We experience autocorrelation when
0)( ≠jiuuE

1. Introduction
Sometimes the term autocorrelation is used
interchangeably.
However, some authors prefer to distinguish between
them.
For example, Tintner defines autocorrelation as ‘lag
correlation of a given series within itself, lagged by a
number of times units’ whereas serial correlation is the
‘lag correlation between two different series’.
We will use both term simultaneously in this lecture.

1. Introduction
There are different types of serial correlation.
With first-order serial correlation, errors in
one time period are correlated directly with
errors in the ensuing time period.
With positive serial correlation, errors in one
time period are positively correlated with errors
in the next time period.

2. Causes of Autocorrelation
 Inertia - Macroeconomics data experience
cycles/business cycles.
 Specification Bias- Excluded variable
Appropriate equation:
Estimated equation
Estimating the second equation implies
ttttt uXXXY ++++= 4433221 ββββ
tttt vXXY +++= 33221 βββ
ttt uXv += 44β

 Specification Bias- Incorrect Functional
Form
tttt vXXY +++= 2
23221 βββ
ttt uXY ++= 221 ββ
ttt vXu += 2
23β

 Cobweb Phenomenon
In agricultural market, the supply reacts to
price with a lag of one time period because
supply decisions take time to implement. This
is known as the cobweb phenomenon.
Thus, at the beginning of this year’s planting
of crops, farmers are influenced by the price
prevailing last year.

 Lags
The above equation is known as auto
regression because one of the explanatory
variables is the lagged value of the dependent
variable.
If you neglect the lagged the resulting error
term will reflect a systematic pattern due to the
influence of lagged consumption on current
consumption.
ttt unConsumptionConsumptio ++= −121 ββ

 Data Manipulation
This equation is known as the first difference form
and dynamic regression model. The previous
equation is known as the level form.
Note that the error term in the first equation is not
auto correlated but it can be shown that the error
term in the first difference form is auto correlated.
ttt uXY ++= 21 ββ 11211 −−−
++= ttt uXY ββ
ttt vXY +∆=∆ 2β

 Nonstationarity
When dealing with time series data, we
should check whether the given time series is
stationary.
A time series is stationary if its characteristics
(e.g. mean, variance and covariance) are time
variant; that is, they do not change over time.
If that is not the case, we have a non
stationary time series.

Suppose Yt is related to X2t and X3t, but we
wrongfully do not include X3t in our model.
The effect of X3t will be captured by the
disturbances ut. If X3t like many economic series
exhibit a trend over time, then X3t depends on
X3t-1,X3t -2and so on. Similarly then ut depends
on ut-1, ut-2 and so on.

Suppose Yt is related to X2t with a quadratic
relationship:
Yt=β1+β2X22t+ut
but we wrongfully assume and estimate a straight
line:
Yt=β1+β2X2t+ut
Then the error term obtained from the straight line
will depend on X22t.

Suppose a company updates its inventory at a
given period in time.
If a systematic error occurred then the cumulative
inventory stock will exhibit accumulated
measurement errors.
These errors will show up as an auto correlated
procedure.

The simplest and most commonly observed is the
first-order autocorrelation.
Consider the multiple regression model:
Yt=β1+β2X2t+β3X3t+β4X4t+…+βkXkt+ut
in which the current observation of the error term
ut is a function of the previous (lagged)
observation of the error term:
ut=ρut-1+et

The coefficient ρis called the first-order
autocorrelation coefficient and takes values from
-1 to +1.
It is obvious that the size of ρ will determine
the strength of serial correlation.
We can have three different cases

If ρ is zero, then we have no autocorrelation.
If ρ approaches unity, the value of the previous
observation of the error becomes more important
in determining the value of the current error and
therefore high degree of autocorrelation exists. In
this case we have positive autocorrelation.
If ρ approaches -1, we have high degree of
negative autocorrelation.

Second-order when:
ut=ρ1ut-1+ ρ2ut-2+et
Third-order when
ut=ρ1ut-1+ ρ2ut-2+ρ3ut-3 +et
p-th order when:
ut=ρ1ut-1+ ρ2ut-2+ρ3ut-3 +…+ ρput-p +et

 The OLS estimators are still unbiased and consistent.
This is because both unbiasedness and consistency do
not depend on assumption 6 which is in this case violated.
 The OLS estimators will be inefficient and therefore no
longer BLUE.
 The estimated variances of the regression coefficients will
be biased and inconsistent, and therefore hypothesis
testing is no longer valid. In most of the cases, the R2
will
be overestimated and the t-statistics will tend to be higher.

There are two ways in general.
The first is the informal way which is done
through graphs and therefore we call it the
graphical method.
The second is through formal tests for
autocorrelation, like the following ones:
 The Durbin Watson Test
 The Breusch-Godfrey Test
 Run test

The following assumptions should be
satisfied:
 The regression model includes a constant
 Autocorrelation is assumed to be of first-order
only
 The equation does not include a lagged
dependent variable as an explanatory variable

 Step 1: Estimate the model by OLS and obtain the
residuals
 Step 2: Calculate the DW statistic
 Step 3: Construct the table with the calculated DW
statistic and the dU, dL, 4-dU and 4-dL critical
values.
 Step 4: Conclude

It is a Lagrange Multiplier Test that resolves the
drawbacks of the DW test.
 Consider the model:
where:
ut=ρ1ut-1+ ρ2ut-2+ρ3ut-3 +…+ ρput-p +et
 Combining those two we get:
Yt=β1+β2X2t+β3X3t+β4X4t+…+βkXkt+
+ρ1ut-1+ ρ2ut-2+ρ3ut-3 +…+ ρput-p +et
Topic NineTopic NineSerial CorrelationSerial Correlation

The null and the alternative hypotheses are:
 H0:ρ1= ρ2=…= ρp=0 no autocorrelation
 Ha:at least one of the ρ’s is not zero, thus, autocorrelation
Step 1: Estimate the model and obtain the residuals
Step 2: Run the full LM model with the number of lags used
being determined by the assumed order of autocorrelation.
Step 3: Compute the LM statistic = (n-ρ)R2
from the LM model
and compare it with the chi-square critical value.
Step 4: Conclude

We have two different cases:
 When ρ is known
 When ρ is unknown
Consider the model
where
ut=ρ1ut-1+et

 Write the model of t-1:
Yt-1=β1+β2X2t-1+β3X3t-1+β4X4t-1+…+βkXkt-1+ut-1
Multiply both sides by ρto get
ρYt-1= ρβ1+ ρβ2X2t-1+ ρβ3X3t-1+ ρβ4X4t-1
+…+ ρβkXkt-1+ ρut-1
 Subtract those two equations:
Yt-ρYt-1= (1-ρ)β1+ β2(X2t-ρX2t-1)+ β3(X3t-ρX3t-1)+
+…+ βk(Xkt-ρXkt-1)+(ut-ρut-1)
or
Y*t= β*1+ β*2X*2t+ β*3X*3t+…+ β*kX*kt+et

Where now the problem of autocorrelation is
resolved because et is no longer autocorrelated.
Note that because from the transformation
we lose one observation, in order to avoid that
loss we generate Y1 and Xi1 as follows:
Y*1=Y1 sqrt(1- ρ2
)
X*i1=Xi1 sqrt(1-ρ2
)
This transformation is known as the quasi-
differencing or generalised differencing.

 The Cochrane-Orcutt iterative procedure.
Step 1: Estimate the regression and obtain
residuals
Step 2: Estimate ρ from regressing the residuals
to its lagged terms.
Step 3: Transform the original variables as
starred variables using the obtained from step 2.

Step 4: Run the regression again with the
transformed variables and obtain residuals.
Step 5 and on: Continue repeating steps 2 to 4
for several rounds until (stopping rule) the
estimates of from two successive iterations differ
by no more than some preselected small value,
such as 0.001.

MUHAMMAD IRFAN HUSSIAN(39)
DR ANWAR UL HAQ ( 28)
AMMER UMER (MPA) (30)
CH AAMAR LASHARI (45)

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YouYou
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Auto Correlation Presentation

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Auto Correlation Presentation