Chapter 3
A brief overview of the
classical linear regression model
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Regression
• Regression is probably the single most important tool at the
econometrician’s disposal.
But what is regression analysis?
• It is concerned with describing and evaluating the relationship between
a given variable (usually called the dependent variable) and one or
more other variables (usually known as the independent variable(s)).
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Some Notation
• Denote the dependent variable by y and the independent variable(s) by x1, x2,
... , xk where there are k independent variables.
• Some alternative names for the y and x variables:
y
x
dependent variable
independent variables
regressand
regressors
effect variable
causal variables
explained variable
explanatory variable
• Note that there can be many x variables but we will limit ourselves to the
case where there is only one x variable to start with. In our set-up, there is
only one y variable.
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Regression is different from Correlation
• If we say y and x are correlated, it means that we are treating y and x in
a completely symmetrical way.
• In regression, we treat the dependent variable (y) and the independent
variable(s) (x’s) very differently. The y variable is assumed to be
random or “stochastic” in some way, i.e. to have a probability
distribution. The x variables are, however, assumed to have fixed
(“non-stochastic”) values in repeated samples.
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Simple Regression
• For simplicity, say k=1. This is the situation where y depends on only one x
variable.
• Examples of the kind of relationship that may be of interest include:
– How asset returns vary with their level of market risk
– Measuring the long-term relationship between stock prices and
dividends.
– Constructing an optimal hedge ratio
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Simple Regression: An Example
• Suppose that we have the following data on the excess returns on a fund
manager’s portfolio (“fund XXX”) together with the excess returns on a
market index:
Year, t
1
2
3
4
5
Excess return
= rXXX,t – rft
17.8
39.0
12.8
24.2
17.2
Excess return on market index
= rmt - rft
13.7
23.2
6.9
16.8
12.3
• We have some intuition that the beta on this fund is positive, and we
therefore want to find whether there appears to be a relationship between
x and y given the data that we have. The first stage would be to form a
scatter plot of the two variables.
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Graph (Scatter Diagram)
Excess return on fund XXX
45
40
35
30
25
20
15
10
5
0
0
5
10
15
20
25
Excess return on market portfolio
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Finding a Line of Best Fit
• We can use the general equation for a straight line,
y=a+bx
to get the line that best “fits” the data.
• However, this equation (y=a+bx) is completely deterministic.
• Is this realistic? No. So what we do is to add a random disturbance
term, u into the equation.
yt =  + xt + ut
where t = 1,2,3,4,5
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Why do we include a Disturbance term?
• The disturbance term can capture a number of features:
- We always leave out some determinants of yt
- There may be errors in the measurement of yt that cannot be
modelled.
- Random outside influences on yt which we cannot model
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Determining the Regression Coefficients
• So how do we determine what  and  are?
• Choose  and  so that the (vertical) distances from the data points to the
fitted lines are minimised (so that the line fits the data as closely as
y
possible):
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Ordinary Least Squares
• The most common method used to fit a line to the data is known as
OLS (ordinary least squares).
• What we actually do is take each distance and square it (i.e. take the
area of each of the squares in the diagram) and minimise the total sum
of the squares (hence least squares).
• Tightening up the notation, let
yt denote the actual data point t
yˆt denote the fitted value from the regression line
uˆ t denote the residual, yt - yˆt
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Actual and Fitted Value
y
yi
uˆ i
yˆ i
xi
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How OLS Works
5
• So min. uˆ1  uˆ2  uˆ3  uˆ4  uˆ5 , or minimise
as the residual sum of squares.
2
2
2
2
2
 uˆ
t 1
2
t
. This is known
• But what was uˆ t ? It was the difference between the actual point and
the line, yt - yˆt .
2
ˆ


y

y
t
• So minimising  t
is equivalent to minimising
with respect to
2
ˆ
u
t
$ and $ .
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Deriving the OLS Estimator
• But y
ˆt  ˆ  ˆxt , so let
L   ( yt  yˆ t ) 2   ( yt  ˆ  ˆxt ) 2
t
• Want to minimise L with respect to (w.r.t.)
and $
$
i
and
$
, so
$ differentiate L w.r.t.
L
 2 ( yt  ˆ  ˆxt )  0
ˆ
t
L
 2 xt ( yt  ˆ  ˆxt )  0
t
ˆ
• From (1),
(1)
(2)
 ( y t  ˆ  ˆxt )  0  y t  Tˆ  ˆ  xt  0
t
• But  yt  Ty and  xt  Tx .
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Deriving the OLS Estimator (cont’d)
• So we can write Ty  Tˆ  Tˆx  0 or
• From (2),  x ( y  ˆ  ˆx )  0
t
t
y  ˆ  ˆx  0
t
(3)
(4)
t
• From (3), ˆ  y  ˆx
• Substitute into (4) for
(5)
$ from (5),
 xt ( yt  y  ˆx  ˆxt )  0
t
2
ˆ
ˆ
x
y

y
x


x
x


x
 t t  t
 t
 t 0
t
2
2
ˆ
ˆ
x
y

T
y
x


T
x


x
 t t
 t 0
t
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Deriving the OLS Estimator (cont’d)
• Rearranging for $ ,
ˆ (Tx 2   xt2 )  Tyx   xt yt
• So overall we have
xt yt  Tx y

ˆ
ˆx
ˆ

and


y


 xt2  Tx 2
• This method of finding the optimum is known as ordinary least squares.
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What do We Use
$ and $ For?
• In the CAPM example used above, plugging the 5 observations in to make up
the formulae given above would lead to the estimates
$ = -1.74 and $= 1.64. We would write the fitted line as:
yˆ t  1.74  1.64x t
• Question: If an analyst tells you that she expects the market to yield a return
20% higher than the risk-free rate next year, what would you expect the return
on fund XXX to be?
• Solution: We can say that the expected value of y = “-1.74 + 1.64 * value of x”,
so plug x = 20 into the equation to get the expected value for y:
yˆ i  1.74  1.6420  31.06
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Accuracy of Intercept Estimate
• Care needs to be exercised when considering the intercept estimate,
particularly if there are no or few observations close to the y-axis:
y
0
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The Population and the Sample
• The population is the total collection of all objects or people to be studied,
for example,
• Interested in
predicting outcome
of an election
Population of interest
the entire electorate
• A sample is a selection of just some items from the population.
• A random sample is a sample in which each individual item in the
population is equally likely to be drawn.
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The DGP and the PRF
• The population regression function (PRF) is a description of the model that
is thought to be generating the actual data and the true relationship
between the variables (i.e. the true values of  and ).
• The PRF is
yt    xt  ut
• The SRF is yˆt  ˆ  ˆxt
and we also know that uˆt  yt  yˆt.
• We use the SRF to infer likely values of the PRF.
• We also want to know how “good” our estimates of  and  are.
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Linearity
• In order to use OLS, we need a model which is linear in the parameters (
and  ). It does not necessarily have to be linear in the variables (y and x).
• Linear in the parameters means that the parameters are not multiplied
together, divided, squared or cubed etc.
• Some models can be transformed to linear ones by a suitable substitution
or manipulation, e.g. the exponential regression model
Yt  e X t eut ln Yt     ln X t  ut
• Then let yt=ln Yt and xt=ln Xt
yt    xt  ut
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Linear and Non-linear Models
• This is known as the exponential regression model. Here, the coefficients
can be interpreted as elasticities.
• Similarly, if theory suggests that y and x should be inversely related:
yt   

xt
 ut
then the regression can be estimated using OLS by substituting
1
zt 
xt
• But some models are intrinsically non-linear, e.g.

yt    xt  ut
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Estimator or Estimate?
• Estimators are the formulae used to calculate the coefficients
• Estimates are the actual numerical values for the coefficients.
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The Assumptions Underlying the
Classical Linear Regression Model (CLRM)
• The model which we have used is known as the classical linear regression model.
• We observe data for xt, but since yt also depends on ut, we must be specific about
how the ut are generated.
• We usually make the following set of assumptions about the ut’s (the unobservable
error terms):
• Technical Notation
Interpretation
1. E(ut) = 0
The errors have zero mean
2. Var (ut) = 2
The variance of the errors is constant and finite
over all values of xt
3. Cov (ui,uj)=0
The errors are statistically independent of
one another
4. Cov (ut,xt)=0
No relationship between the error and
corresponding x variate
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The Assumptions Underlying the
CLRM Again
• An alternative assumption to 4., which is slightly stronger, is that the
xt’s are non-stochastic or fixed in repeated samples.
• A fifth assumption is required if we want to make inferences about the
population parameters (the actual  and ) from the sample parameters
( $ and $ )
• Additional Assumption
5. ut is normally distributed
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Properties of the OLS Estimator
• If assumptions 1. through 4. hold, then the estimators $ and$ determined by
OLS are known as Best Linear Unbiased Estimators (BLUE).
What does the acronym stand for?
• “Estimator”
• “Linear”
• “Unbiased”
• “Best”
- $ is an estimator of the true value of .
- $ is a linear estimator
- On average, the actual value of the $ and $’s will be equal to
the true values.
- means that the OLS estimator $ has minimum variance among
the class of linear unbiased estimators. The Gauss-Markov
theorem proves that the OLS estimator is best.
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Consistency/Unbiasedness/Efficiency
• Consistent
The least squares estimators $ and $ are consistent. That is, the estimates will
converge to their true values as the sample size increases to infinity. Need
the
assumptions E(xtut)=0 and Var(ut)=2 <  to prove this. Consistency implies that
lim Pr ˆ      0   0
T 
• Unbiased
The least squares estimates of $ and $ are unbiased. That is E($)= and E($ )=
Thus on average the estimated value will be equal to the true values. To prove
this also requires the assumption that E(ut)=0. Unbiasedness is a stronger
condition than consistency.
• Efficiency
An estimator $ of parameter  is said to be efficient if it is unbiased and no other
unbiased estimator has a smaller variance. If the estimator is efficient, we are
minimising the probability that it is a long way off from the true value of .


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Precision and Standard Errors
• Any set of regression estimates of $ and $ are specific to the sample used in
their estimation.
• Recall that the estimators of  and  from the sample parameters ($ and $) are
given by
x y  Tx y
ˆ   t 2 t
andˆ  y  ˆx
2
 xt  Tx
• What we need is some measure of the reliability or precision of the estimators
( $ and $ ). The precision of the estimate is given by its standard error. Given
assumptions 1 - 4 above, then the standard errors can be shown to be given by
2
2
x
x
 t ,
 t s
SE(ˆ )  s
T  ( xt  x ) 2
T  xt2  T 2 x 2
SE( ˆ )  s
1
s
2
 ( xt  x )
1
2
2
x

T
x
 t
where s is the estimated standard deviation of the residuals.
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Estimating the Variance of the Disturbance Term
• The variance of the random variable ut is given by
Var(ut) = E[(ut)-E(ut)]2
which reduces to
Var(ut) = E(ut2)
• We could estimate this using the average of ut2:
s2 
1
ut2

T
• Unfortunately this is not workable since ut is not observable. We can use
the sample counterpart to ut, which is uˆ t :
1
2
2
s 
But this estimator is a biased estimator of 2.
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uˆ

T
t
29
Estimating the Variance of the Disturbance Term
(cont’d)
• An unbiased estimator of  is given by
where
uˆ
2
t
s
2
ˆ
u
t
T 2
is the residual sum of squares and T is the sample size.
Some Comments on the Standard Error Estimators
1. Both SE($ ) and SE($ ) depend on s2 (or s). The greater the variance s2, then
the more dispersed the errors are about their mean value and therefore the
more dispersed y will be about its mean value.
2. The sum of the squares of x about their mean appears in both formulae.
The larger the sum of squares, the smaller the coefficient variances.
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Some Comments on the Standard Error Estimators
Consider what happens if
2


x

x
is small or large:
 t
y
y
y
y
x
0
x
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0
x
x
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Some Comments on the Standard Error Estimators
(cont’d)
3. The larger the sample size, T, the smaller will be the coefficient
variances. T appears explicitly in SE($ ) and implicitly in SE( $ ).
2


x

x
T appears implicitly since the sum  t
is from t = 1 to T.
2
4. The term  xt appears in the SE($ ).
2
The reason is that  xt measures how far the points are away from the
y-axis.
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Example: How to Calculate the Parameters and
Standard Errors
• Assume we have the following data calculated from a regression of y on a
single variable x and a constant over 22 observations.
• Data:
 xt yt  830102, T  22, x  416.5, y  86.65,
2
x
 t  3919654, RSS  130.6
830102  (22 * 416.5 * 86.65)
• Calculations: $ 
2  0.35
3919654  22 *(416.5)
$  86.65  035
. * 4165
.  5912
.
• We write
yˆt  ˆ  ˆxt
yˆt  59.12  0.35xt
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Example (cont’d)
uˆ t2
130.6

• SE(regression), s 

 2.55
T 2
20
SE( )  2.55*
3919654
 3.35
2
22 3919654  22 416.5
SE(  )  2.55*
1
 0.0079
2
3919654 22 416.5




• We now write the results as
yˆ t   59.12  0.35xt
(3.35) (0.0079)
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An Introduction to Statistical Inference
• We want to make inferences about the likely population values from
the regression parameters.
Example: Suppose we have the following regression results:
yˆ t  20.3  0.5091xt
(14.38) (0.2561)
• $  0.5091 is a single (point) estimate of the unknown population
parameter, . How “reliable” is this estimate?
• The reliability of the point estimate is measured by the coefficient’s
standard error.
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Hypothesis Testing: Some Concepts
• We can use the information in the sample to make inferences about the
population.
• We will always have two hypotheses that go together, the null hypothesis
(denoted H0) and the alternative hypothesis (denoted H1).
• The null hypothesis is the statement or the statistical hypothesis that is actually
being tested. The alternative hypothesis represents the remaining outcomes of
interest.
• For example, suppose given the regression results above, we are interested in
the hypothesis that the true value of  is in fact 0.5. We would use the notation
H0 :  = 0.5
H1 :   0.5
This would be known as a two sided test.
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One-Sided Hypothesis Tests
• Sometimes we may have some prior information that, for example, we
would expect  > 0.5 rather than  < 0.5. In this case, we would do a
one-sided test:
H0 :  = 0.5
H1 :  > 0.5
or we could have had
H0 :  = 0.5
H1 :  < 0.5
• There are two ways to conduct a hypothesis test: via the test of
significance approach or via the confidence interval approach.
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The Probability Distribution of the
Least Squares Estimators
• We assume that ut  N(0,2)
• Since the least squares estimators are linear combinations of the random
variables
i.e. $   wt yt
• The weighted sum of normal random variables is also normally distributed, so
$  N(, Var())
$  N(, Var())
• What if the errors are not normally distributed? Will the parameter estimates
still be normally distributed?
• Yes, if the other assumptions of the CLRM hold, and the sample size is
sufficiently large.
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The Probability Distribution of the
Least Squares Estimators (cont’d)
• Standard normal variates can be constructed from $ and $ :
ˆ  
~ N 0,1 and
var 
ˆ  
~ N 0,1
var  
• But var() and var() are unknown, so
ˆ  
~ tT 2 and
SE(ˆ )
ˆ  
~ tT 2
ˆ
SE(  )
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Testing Hypotheses:
The Test of Significance Approach
• Assume the regression equation is given by ,
for t=1,2,...,T
yt    xt  ut
• The steps involved in doing a test of significance are:
1. Estimate $ , $ and SE($ ) , SE( $ ) in the usual way
2. Calculate the test statistic. This is given by the formula
$   *
test statistic 
SE ( $ )
where  * is the value of  under the null hypothesis.
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The Test of Significance Approach (cont’d)
3. We need some tabulated distribution with which to compare the estimated
test statistics. Test statistics derived in this way can be shown to follow a tdistribution with T-2 degrees of freedom.
As the number of degrees of freedom increases, we need to be less cautious in
our approach since we can be more sure that our results are robust.
4. We need to choose a “significance level”, often denoted . This is also
sometimes called the size of the test and it determines the region where we
will reject or not reject the null hypothesis that we are testing. It is
conventional to use a significance level of 5%.
Intuitive explanation is that we would only expect a result as extreme as this
or more extreme 5% of the time as a consequence of chance alone.
Conventional to use a 5% size of test, but 10% and 1% are also commonly
used.
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Determining the Rejection Region for a Test of
Significance
5. Given a significance level, we can determine a rejection region and nonrejection region. For a 2-sided test:
f(x)
2.5%
rejection region
95% non-rejection
region
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2.5%
rejection region
42
The Rejection Region for a 1-Sided Test (Upper Tail)
f(x)
95% non-rejection
region
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5% rejection region
43
The Rejection Region for a 1-Sided Test (Lower Tail)
f(x)
95% non-rejection region
5% rejection region
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The Test of Significance Approach: Drawing
Conclusions
6. Use the t-tables to obtain a critical value or values with which to
compare the test statistic.
7. Finally perform the test. If the test statistic lies in the rejection
region then reject the null hypothesis (H0), else do not reject H0.
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A Note on the t and the Normal Distribution
• You should all be familiar with the normal distribution and its
characteristic “bell” shape.
• We can scale a normal variate to have zero mean and unit variance by
subtracting its mean and dividing by its standard deviation.
• There is, however, a specific relationship between the t- and the
standard normal distribution. Both are symmetrical and centred on
zero. The t-distribution has another parameter, its degrees of freedom.
We will always know this (for the time being from the number of
observations -2).
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What Does the t-Distribution Look Like?
normal distribution
t-distribution
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Comparing the t and the Normal Distribution
• In the limit, a t-distribution with an infinite number of degrees of freedom is
a standard normal, i.e. t ()  N (01
,)
• Examples from statistical tables:
Significance level
N(0,1)
50%
0
5%
1.64
2.5%
1.96
0.5%
2.57
t(40)
0
1.68
2.02
2.70
t(4)
0
2.13
2.78
4.60
• The reason for using the t-distribution rather than the standard normal is that
we had to estimate  2, the variance of the disturbances.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
48
The Confidence Interval Approach
to Hypothesis Testing
• An example of its usage: We estimate a parameter, say to be 0.93, and
a “95% confidence interval” to be (0.77,1.09). This means that we are
95% confident that the interval containing the true (but unknown)
value of .
• Confidence intervals are almost invariably two-sided, although in
theory a one-sided interval can be constructed.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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How to Carry out a Hypothesis Test
Using Confidence Intervals
1. Calculate $ , $ and SE($ ) , SE( $ ) as before.
2. Choose a significance level, , (again the convention is 5%). This is equivalent to
choosing a (1-)100% confidence interval, i.e. 5% significance level = 95%
confidence interval
3. Use the t-tables to find the appropriate critical value, which will again have T-2
degrees of freedom.
4. The confidence interval is given by
( ˆ  t crit  SE( ˆ ), ˆ  t crit  SE( ˆ ))
5. Perform the test: If the hypothesised value of  (*) lies outside the confidence
interval, then reject the null hypothesis that  = *, otherwise do not reject the null.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
50
Confidence Intervals Versus Tests of Significance
• Note that the Test of Significance and Confidence Interval approaches
always give the same answer.
• Under the test of significance approach, we would not reject H0 that  = *
if the test statistic lies within the non-rejection region, i.e. if
$   *
tcrit £
£ tcrit
$
SE (  )
• Rearranging, we would not reject if
 t crit  SE ( ˆ ) £ ˆ   * £ t crit  SE ( ˆ )
ˆ  t crit  SE( ˆ ) £  * £ ˆ  t crit  SE( ˆ )
• But this is just the rule under the confidence interval approach.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Constructing Tests of Significance and
Confidence Intervals: An Example
• Using the regression results above,
yˆ t  20.3  0.5091xt , T=22
(14.38) (0.2561)
• Using both the test of significance and confidence interval approaches,
test the hypothesis that  =1 against a two-sided alternative.
• The first step is to obtain the critical value. We want tcrit = t20;5%
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
52
Determining the Rejection Region
f(x)
2.5% rejection region
-2.086
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
2.5% rejection region
+2.086
53
Performing the Test
• The hypotheses are:
H0 :  = 1
H1 :   1
Test of significance
approach
$   *
test stat 
SE ( $ )
05091
.
1

 1917
.
0.2561
Do not reject H0 since
test stat lies within
non-rejection region
Confidence interval
approach
ˆ  t crit  SE ( ˆ )
 0.5091  2.086  0.2561
 (0.0251,1.0433)
Since 1 lies within the
confidence interval,
do not reject H0
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
54
Testing other Hypotheses
• What if we wanted to test H0 :  = 0 or H0 :  = 2?
• Note that we can test these with the confidence interval approach.
For interest (!), test
H0 :  = 0
vs. H1 :   0
vs.
H0 :  = 2
H1 :   2
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
55
Changing the Size of the Test
• But note that we looked at only a 5% size of test. In marginal cases
(e.g. H0 :  = 1), we may get a completely different answer if we use a
different size of test. This is where the test of significance approach is
better than a confidence interval.
• For example, say we wanted to use a 10% size of test. Using the test of
significance approach,
$   *
test stat 
SE ( $ )
05091
.
1

 1917
.
0.2561
as above. The only thing that changes is the critical t-value.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Changing the Size of the Test:
The New Rejection Regions
f(x)
5% rejection region
-1.725
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
5% rejection region
+1.725
57
Changing the Size of the Test:
The Conclusion
• t20;10% = 1.725. So now, as the test statistic lies in the rejection region,
we would reject H0.
• Caution should therefore be used when placing emphasis on or making
decisions in marginal cases (i.e. in cases where we only just reject or
not reject).
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
58
Some More Terminology
• If we reject the null hypothesis at the 5% level, we say that the result
of the test is statistically significant.
• Note that a statistically significant result may be of no practical
significance. E.g. if a shipment of cans of beans is expected to weigh
450g per tin, but the actual mean weight of some tins is 449g, the
result may be highly statistically significant but presumably nobody
would care about 1g of beans.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
59
The Errors That We Can Make
Using Hypothesis Tests
• We usually reject H0 if the test statistic is statistically significant at a
chosen significance level.
• There are two possible errors we could make:
1. Rejecting H0 when it was really true. This is called a type I error.
2. Not rejecting H0 when it was in fact false. This is called a type II error.
Result of
Test
Significant
(reject H0)
Insignificant
( do not
reject H0)
Reality
H0 is true
Type I error
=
H0 is false


Type II error
=
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
60
The Trade-off Between Type I and Type II Errors
• The probability of a type I error is just , the significance level or size of test we
chose. To see this, recall what we said significance at the 5% level meant: it is only
5% likely that a result as or more extreme as this could have occurred purely by
chance.
• Note that there is no chance for a free lunch here! What happens if we reduce the size
of the test (e.g. from a 5% test to a 1% test)? We reduce the chances of making a type
I error ... but we also reduce the probability that we will reject the null hypothesis at
all, so we increase the probability of a type II error:
less likely
to falsely reject
Reduce size 
of test
more strict 
criterion for
rejection
reject null
hypothesis
less often
more likely to
incorrectly not
reject
• So there is always a trade off between type I and type II errors when choosing a
significance level. The only way we can reduce the chances of both is to increase the
sample size.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
61
The Exact Significance Level or p-value
• This is equivalent to choosing an infinite number of critical t-values from
tables. It gives us the marginal significance level where we would be
indifferent between rejecting and not rejecting the null hypothesis.
• If the test statistic is large in absolute value, the p-value will be small, and
vice versa. The p-value gives the plausibility of the null hypothesis.
e.g. a test statistic is distributed as a t62 = 1.47.
The p-value = 0.12.
• Do we reject at the 5% level?...........................No
• Do we reject at the 10% level?.........................No
• Do we reject at the 20% level?.........................Yes
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Generalising the Simple Model to
Multiple Linear Regression
• Before, we have used the model
yt    xt  ut t = 1,2,...,T
• But what if our dependent (y) variable depends on more than one
independent variable?
For example the number of cars sold might plausibly depend on
1. the price of cars
2. the price of public transport
3. the price of petrol
4. the extent of the public’s concern about global warming
• Similarly, stock returns might depend on several factors.
• Having just one independent variable is no good in this case - we want to
have more than one x variable. It is very easy to generalise the simple
model to one with k-1 regressors (independent variables).
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
63
Multiple Regression and the Constant Term
• Now we write
yt  1   2 x2t  3 x3t  ...  k xkt  ut , t=1,2,...,T
• Where is x1? It is the constant term. In fact the constant term is usually
represented by a column of ones of length T:
1
1
x1   


1
1 is the coefficient attached to the constant term (which we called  before).
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
64
Different Ways of Expressing
the Multiple Linear Regression Model
• We could write out a separate equation for every value of t:
y1  1   2 x21   3 x31  ...   k xk1  u1
y2  1   2 x22   3 x32  ...   k xk 2  u2



yT  1   2 x2T   3 x3T  ...   k xkT  uT
• We can write this in matrix form
y = X +u
where
y is T  1
X is T  k
 is k  1
u is T  1
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
65
Inside the Matrices of the
Multiple Linear Regression Model
• e.g. if k is 2, we have 2 regressors, one of which is a column of ones:
 y1  1 x21 
 u1 
 y  1 x  
u 


22 
1
 2  
 2

       2    
  

 
 yT  1 x2T 
uT 
T 1
T2
21
T1
• Notice that the matrices written in this way are conformable.
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66
How Do We Calculate the Parameters (the  )
in this Generalised Case?
• Previously, we took the residual sum of squares, and minimised it w.r.t.
 and .
• In the matrix notation, we have
 uˆ 1 
 uˆ 
uˆ   2 
  
 
uˆ T 
• The RSS would be given by
uˆ ' uˆ  uˆ1 uˆ2
 uˆ1 
uˆ 
 uˆT  2   uˆ12  uˆ22  ...  uˆT2   uˆt2
 
 
uˆT 
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
67
The OLS Estimator for the
Multiple Regression Model
• In order to obtain the parameter estimates, 1, 2,..., k, we would
minimise the RSS with respect to all the s.
• It can be shown that
 ˆ1 
 
 ˆ 2 
ˆ
     ( X X ) 1 X  y

 
 ˆ k 
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
68
Calculating the Standard Errors for the
Multiple Regression Model
• Check the dimensions: $ is k  1 as required.
• But how do we calculate the standard errors of the coefficient estimates?
• Previously, to estimate the variance of the errors, 2, we used s 2
•
uˆ


2
t
T 2
.
u$' u$
2
Now using the matrix notation, we use s 
Tk
• where k = number of regressors. It can be proved that the OLS estimator of
the variance of $ is given by the diagonal elements of s2 ( X ' X )1 , so that
the variance of $1 is the first element, the variance of $2 is the second
element, and …, and the variance of $k is the kth diagonal element.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
69
Calculating Parameter and Standard Error Estimates
for Multiple Regression Models: An Example
• Example: The following model with k=3 is estimated over 15 observations:
y  1   2 x2  3 x3  u
and the following data have been calculated from the original X’s.
•
•
 2.0 35
30
. 10
. 
. 
( X ' X ) 1   35
. 10
. 65
.  ,( X ' y)   2.2  , u$' u$  10.96
10
 0.6 
. 65
. 4.3 
Calculate the coefficient estimates and their standard errors.
To calculate the coefficients, just multiply the matrix by the vector to obtain
X ' X 1 X ' y .
To calculate the standard errors, we need an estimate of 2.
s2 
RSS 10.96

 091
.
T  k 15  3
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
70
Calculating Parameter and Standard Error Estimates
for Multiple Regression Models: An Example (cont’d)
• The variance-covariance matrix of $ is given by
 183
.
320
.
0.91
s2 ( X ' X ) 1  0.91( X ' X ) 1   320
.
0.91 594
. 
0.91 594
.
393
. 
• The variances are on the leading diagonal:
Var ( $1 )  183
.
SE ( $1 )  135
.
Var ( $2 )  0.91  SE ( $2 )  0.96
Var ( $ )  3.93
SE ( $ )  198
.
3
3
• We write: yˆ  1.10  4.40x2t  19.88x3t
1.35 0.96 1.98
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
71
A Special Type of Hypothesis Test: The t-ratio
• Recall that the formula for a test of significance approach to hypothesis
testing using a t-test was
$i   i*
test statistic 
SE $i 
H 0 : i = 0
H 1 : i  0
i.e. a test that the population coefficient is zero against a two-sided
alternative, this is known as a t-ratio test:
• If the test is
$i
Since  i* = 0, test stat 
SE ( $i )
• The ratio of the coefficient to its SE is known as the t-ratio or t-statistic.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
72
The t-ratio: An Example
• In the last example above:
Coefficient
SE
t-ratio
1.10
1.35
0.81
Compare this with a tcrit with 15-3
(2½% in each tail for a 5% test)
Do we reject H0:
H0:
H0:
1 = 0?
2 = 0?
3 = 0?
-4.40
0.96
-4.63
=
=
=
12 d.f.
2.179
3.055
19.88
1.98
10.04
5%
1%
(No)
(Yes)
(Yes)
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
73
What Does the t-ratio tell us?
• If we reject H0, we say that the result is significant. If the coefficient is not
“significant” (e.g. the intercept coefficient in the last regression above), then
it means that the variable is not helping to explain variations in y. Variables
that are not significant are usually removed from the regression model.
• In practice there are good statistical reasons for always having a constant
y
even if it is not significant. Look at what happens if no intercept is included:
t
xt
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
74
Data Mining
• Data mining is searching many series for statistical relationships
without theoretical justification.
• For example, suppose we generate one dependent variable and twenty
explanatory variables completely randomly and independently of each
other.
• If we regress the dependent variable separately on each independent
variable, on average one slope coefficient will be significant at 5%.
• If data mining occurs, the true significance level will be greater than
the nominal significance level.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
75
An Example of the Use of a Simple t-test to Test a
Theory in Finance
• Testing for the presence and significance of abnormal returns (“Jensen’s
alpha” - Jensen, 1968).
• The Data: Annual Returns on the portfolios of 115 mutual funds from
1945-1964.
• The model:
R jt  R ft   j   j ( Rmt  R ft )  u jt
for j = 1, …, 115
• We are interested in the significance of j.
• The null hypothesis is H0: j = 0 .
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
76
Frequency Distribution of t-ratios of Mutual Fund
Alphas (gross of transactions costs)
Source Jensen (1968). Reprinted with the permission of Blackwell publishers.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
77
Frequency Distribution of t-ratios of Mutual Fund
Alphas (net of transactions costs)
Source Jensen (1968). Reprinted with the permission of Blackwell publishers.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
78
Can UK Unit Trust Managers “Beat the Market”?
• We now perform a variant on Jensen’s test in the context of the UK market,
considering monthly returns on 76 equity unit trusts. The data cover the
period January 1979 – May 2000 (257 observations for each fund). Some
summary statistics for the funds are:
Mean Minimum Maximum Median
Average monthly return, 1979-2000
1.0%
0.6%
1.4%
1.0%
Standard deviation of returns over time 5.1%
4.3%
6.9%
5.0%
• Jensen Regression Results for UK Unit Trust Returns, January 1979-May
2000
R jt  R ft   j   j ( Rmt  R ft )   jt
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
79
Can UK Unit Trust Managers “Beat the Market”?
: Results
Estimates of


t-ratio on 
Mean
-0.02%
0.91
-0.07
Minimum
-0.54%
0.56
-2.44
Maximum
0.33%
1.09
3.11
Median
-0.03%
0.91
-0.25
• In fact, gross of transactions costs, 9 funds of the sample of 76 were
able to significantly out-perform the market by providing a significant
positive alpha, while 7 funds yielded significant negative alphas.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Performance of UK Unit Trusts 1979-2000
3500
3000
2500
2000
1500
1000
500
0
79
nJa
80
nJa
81
nJa
nJa
82
nJa
83
nJa
84
85
nJa
86
nJa
87
88
89
90
91
92
93
nnnnnnna
a
a
a
a
a
a
J
J
J
J
J
J
J
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
94
nJa
95
nJa
nJa
96
97
nJa
98
nJa
nJa
99
nJa
81
00
The Overreaction Hypothesis and
the UK Stock Market
• Motivation
Two studies by DeBondt and Thaler (1985, 1987) showed that stocks which
experience a poor performance over a 3 to 5 year period tend to outperform
stocks which had previously performed relatively well.
• How Can This be Explained?
2 suggestions
1. A manifestation of the size effect
DeBondt & Thaler did not believe this a sufficient explanation, but Zarowin
(1990) found that allowing for firm size did reduce the subsequent return on
the losers.
2. Reversals reflect changes in equilibrium required returns
Ball & Kothari (1989) find the CAPM beta of losers to be considerably
higher than that of winners.
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82
The Overreaction Hypothesis and
the UK Stock Market (cont’d)
• Another interesting anomaly: the January effect.
– Another possible reason for the superior subsequent performance
of losers.
– Zarowin (1990) finds that 80% of the extra return available from
holding the losers accrues to investors in January.
• Example study: Clare and Thomas (1995)
Data:
Monthly UK stock returns from January 1955 to 1990 on all firms
traded on the London Stock exchange.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
83
Methodology
• Calculate the monthly excess return of the stock over the market over a 12,
24 or 36 month period for each stock i:
Uit = Rit - Rmt
n = 12, 24 or 36 months
• Calculate the average monthly return for the stock i over the first 12, 24, or
36 month period:
1 n
Ri   U it
n t 1
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
84
Portfolio Formation
• Then rank the stocks from highest average return to lowest and from 5
portfolios:
Portfolio 1:
Portfolio 2:
Portfolio 3:
Portfolio 4:
Portfolio 5:
Best performing 20% of firms
Next 20%
Next 20%
Next 20%
Worst performing 20% of firms.
• Use the same sample length n to monitor the performance of each
portfolio.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
85
Portfolio Formation and
Portfolio Tracking Periods
• How many samples of length n have we got?
n = 1, 2, or 3 years.
• If n = 1year:
Estimate for year 1
Monitor portfolios for year 2
Estimate for year 3
...
Monitor portfolios for year 36
• So if n = 1, we have 18 INDEPENDENT (non-overlapping) observation /
tracking periods.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
86
Constructing Winner and Loser Returns
• Similarly, n = 2 gives 9 independent periods and n = 3 gives 6 independent
periods.
• Calculate monthly portfolio returns assuming an equal weighting of stocks in
each portfolio.
• Denote the mean return for each month over the 18, 9 or 6 periods for the
winner and loser portfolios respectively as R pW and RpL respectively.
• Define the difference between these as
L
RDt = Rp -
• Then perform the regression
RDt = 1 + t
• Look at the significance of 1.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
R pW .
(Test 1)
87
Allowing for Differences in the Riskiness
of the Winner and Loser Portfolios
• Problem: Significant and positive 1 could be due to higher return being
required on loser stocks due to loser stocks being more risky.
• Solution: Allow for risk differences by regressing against the market risk
premium:
RDt = 2 + (Rmt-Rft) + t
(Test 2)
where
Rmt is the return on the FTA All-share
Rft is the return on a UK government 3 month t-bill.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
88
Is there an Overreaction Effect in the
UK Stock Market? Results
Panel A: All Months
n = 12
0.0033
0.0036
-0.37%
n = 24
0.0011
-0.0003
1.68%
n =36
0.0129
0.0115
1.56%
Coefficient for (3.47): ˆ1
-0.00031
(0.29)
0.0014**
(2.01)
0.0013
(1.55)
Coefficients for (3.48): ˆ 2
-0.00034
(-0.30)
-0.022
(-0.25)
0.00147**
(2.01)
0.010
(0.21)
0.0013*
(1.41)
-0.0025
(-0.06)
-0.0007
(-0.72)
0.0012*
(1.63)
0.0009
(1.05)
Return on Loser
Return on Winner
Implied annualised return difference
ˆ
Panel B: All Months Except January
Coefficient for (3.47): ˆ1
Notes: t-ratios in parentheses; * and ** denote significance at the 10% and 5% levels
respectively. Source: Clare and Thomas (1995). Reprinted with the permission of Blackwell
Publishers.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
89
Are the Losers Small?
• Rank the firms according to size
i = 1  smallest 20% of firms
...
i = 5  largest 20% of firms
• Then also rank firms according to their average return as previously.
j = 1  worst performing 20% of firms
...
j = 5  best performing 20% of firms
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Portfolio Ranking by Performance and Size
• So R ij is the return on the ith sized portfolio and the jth return ranking
portfolio (dropping time subscripts for simplicity)m, so
RD1  R 11  R 51
RD2  R 12  R 52
RD3  R 13  R 53
RD4  R 14  R 54
RD5  R 15  R 55
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
(4)
(5)
(6)
(7)
(8)
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Overreaction versus Small Firm Effects for
a 1-Year Horizon
Panel A: All Months
Regression Equation
(3.57)
(3.58)
0.00109
0.00509
-0.00141
0.00116
(3.59)
0.00583
0.00415
(3.60)
0.00825
0.00473
(3.61)
0.00778
0.00457
0.0025
(0.93)
0.0040
(1.30)
0.0017
(0.80)
0.0035
(1.05)
0.0032
(1.06)
0.0023
(0.83)
ˆ

-0.21
(-0.99)
Panel B: All Months Except January
-0.0011
Coefficient for Test 1: ˆ1
(-0.44)
0.0032
(1.73)
-0.63**
(-4.50)
0.0014
(0.69)
-0.258
(-1.68)
0.0028
(1.12)
-0.625**
(-3.24)
0.026
(1.06)
-0.50**
(-2.63)
0.0035
(1.09)
0.002
(0.94)
0.0027
(0.75)
0.0038
(1.16)
Return on Loser
Return on Winner
Coefficient for Test 1: ˆ1
Coefficients for Test 2: ˆ 2
Notes: t-ratios in parentheses; * and ** denote significance at the 10% and 5% levels respectively.
Source: Clare and Thomas (1995). Reprinted with the permission of Blackwell Publishers.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Overreaction versus Small Firm Effects for
a 2-Year Horizon
Panel A: All Months
Return on Loser
Return on Winner
Coefficient for Test 1: ˆ1
Coefficients for Test 2: ˆ 2
Regression Equation
(3.57)
(3.58)
0.00591
0.0080
0.00218
0.0026
(3.59)
0.00715
0.00197
(3.60)
0.0044
0.0002
(3.61)
0.00708
-0.00057
0.0037
(1.39)
0.0053**
(2.01)
0.0052**
(1.97)
0.0042**
(1.91)
0.008**
(3.03)
0.004
(1.56)
-0.129**
(-2.08)
0.0034
(1.45)
-0.167**
(-2.90)
0.0035
(1.54)
-0.075
(-1.37)
0.0068**
(2.64)
-0.09
(-1.40)
0.0054*
(1.92)
0.005*
(1.86)
0.0037
(1.60)
0.009**
(3.42)
0.0035
(1.23)
ˆ

-0.024
(-0.34)
Panel B: All Months Except January
0.0028
Coefficient for Test 1: ˆ1
(0.99)
Notes: t-ratios in parentheses; * and ** denote significance at the 10% and 5% levels respectively.
Source: Clare and Thomas (1995). Reprinted with the permission of Blackwell Publishers.
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Overreaction versus Small Firm Effects for
a 3-Year Horizon
Panel A: All Months
Return on Loser
Return on Winner
Coefficient for Test 1: ˆ1
Coefficients for Test 2: ˆ 2
Regression Equation
(3.57)
(3.58)
0.01591
0.01484
0.01155
0.01335
(3.59)
0.01477
0.01176
(3.60)
0.01202
0.01130
(3.61)
0.12420
0.00856
0.0044
(1.55)
0.003
(1.21)
0.0007
(0.21)
0.0039
(1.13)
0.0046* 0.0054**
(1.87)
(2.21)
-0.30** -0.23**
(-3.63)
(-2.77)
0.005
(1.45)
-0.38**
(-3.50)
0.0071**
(2.14)
-0.32**
(-2.81)
0.001
(0.37)
0.00004 0.003
(0.01)
(0.94)
0.0071**
(2.56)
ˆ

-0.26**
(-2.80)
Panel B: All Months Except January
0.0046*
Coefficient for Test 1: ˆ1
(1.73)
0.0015
(0.55)
0.0024
(1.01)
Notes: t-ratios in parentheses; * and ** denote significance at the 10% and 5% levels respectively.
Source: Clare and Thomas (1995). Reprinted with the permission of Blackwell Publishers.
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Testing for Seasonal Effects in Overreactions
• Is there evidence that losers out-perform winners more at one time of the
year than another?
• To test this, calculate the difference between the winner & loser portfolios
as previously, RDt , and regress this on 12 month-of-the-year dummies:
12
RDt   i Mi  t
i 1
• Significant out-performance of losers over winners in,
– June (for the 24-month horizon), and
– January, April and October (for the 36-month horizon)
– winners appear to stay significantly as winners in
• March (for the 12-month horizon).
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Conclusions
• Evidence of overreactions in stock returns.
• Losers tend to be small so we can attribute most of the overreaction in the
UK to the size effect.
Comments
• Small samples
• No diagnostic checks of model adequacy
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Testing Multiple Hypotheses: The F-test
• We used the t-test to test single hypotheses, i.e. hypotheses involving only
one coefficient. But what if we want to test more than one coefficient
simultaneously?
• We do this using the F-test. The F-test involves estimating 2 regressions.
• The unrestricted regression is the one in which the coefficients are freely
determined by the data, as we have done before.
• The restricted regression is the one in which the coefficients are restricted,
i.e. the restrictions are imposed on some s.
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The F-test:
Restricted and Unrestricted Regressions
• Example
The general regression is
yt = 1 + 2x2t + 3x3t + 4x4t + ut
(1)
• We want to test the restriction that 3+4 = 1 (we have some hypothesis
from theory which suggests that this would be an interesting hypothesis to
study). The unrestricted regression is (1) above, but what is the restricted
regression?
yt = 1 + 2x2t + 3x3t + 4x4t + ut s.t. 3+4 = 1
• We substitute the restriction (3+4 = 1) into the regression so that it is
automatically imposed on the data.
3+4 = 1  4 = 1- 3
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The F-test: Forming the Restricted Regression
yt = 1 + 2x2t + 3x3t + (1-3)x4t + ut
yt = 1 + 2x2t + 3x3t + x4t - 3x4t + ut
• Gather terms in ’s together and rearrange
(yt - x4t) = 1 + 2x2t + 3(x3t - x4t) + ut
• This is the restricted regression. We actually estimate it by creating two new
variables, call them, say, Pt and Qt.
Pt = yt - x4t
Qt = x3t - x4t
so
Pt = 1 + 2x2t + 3Qt + ut is the restricted regression we actually estimate.
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Calculating the F-Test Statistic
• The test statistic is given by
RRSS  URSS T  k
test statistic 

URSS
m
where URSS = RSS from unrestricted regression
RRSS = RSS from restricted regression
m = number of restrictions
T
= number of observations
k
= number of regressors in unrestricted regression
including a constant in the unrestricted regression (or the total number
of parameters to be estimated).
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The F-Distribution
• The test statistic follows the F-distribution, which has 2 d.f.
parameters.
• The value of the degrees of freedom parameters are m and (T-k)
respectively (the order of the d.f. parameters is important).
• The appropriate critical value will be in column m, row (T-k).
• The F-distribution has only positive values and is not symmetrical. We
therefore only reject the null if the test statistic > critical F-value.
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Determining the Number of Restrictions in an F-test
• Examples :
H0: hypothesis
No. of restrictions, m
1 + 2 = 2
1
2 = 1 and 3 = -1
2
2 = 0, 3 = 0 and 4 = 0
3
• If the model is yt = 1 + 2x2t + 3x3t + 4x4t + ut,
then the null hypothesis
H0: 2 = 0, and 3 = 0 and 4 = 0 is tested by the regression F-statistic. It
tests the null hypothesis that all of the coefficients except the intercept
coefficient are zero.
• Note the form of the alternative hypothesis for all tests when more than one
restriction is involved: H1: 2  0, or 3  0 or 4  0
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What we Cannot Test with Either an F or a t-test
• We cannot test using this framework hypotheses which are not linear
or which are multiplicative, e.g.
H0: 2 3 = 2 or H0: 2 2 = 1
cannot be tested.
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The Relationship between the t and the FDistributions
• Any hypothesis which could be tested with a t-test could have been
tested using an F-test, but not the other way around.
For example, consider the hypothesis
H0: 2 = 0.5
H1: 2  0.5
$2  0.5
We could have tested this using the usual t-test: test stat 
SE ( $2 )
or it could be tested in the framework above for the F-test.
• Note that the two tests always give the same result since the tdistribution is just a special case of the F-distribution.
• For example, if we have some random variable Z, and Z  t (T-k) then
also Z2  F(1,T-k)
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F-test Example
• Question: Suppose a researcher wants to test whether the returns on a company
stock (y) show unit sensitivity to two factors (factor x2 and factor x3) among three
considered. The regression is carried out on 144 monthly observations. The
regression is yt = 1 + 2x2t + 3x3t + 4x4t+ ut
- What are the restricted and unrestricted regressions?
- If the two RSS are 436.1 and 397.2 respectively, perform the test.
• Solution:
Unit sensitivity implies H0:2=1 and 3=1. The unrestricted regression is the one
in the question. The restricted regression is (yt-x2t-x3t)= 1+ 4x4t+ut or letting
zt=yt-x2t-x3t, the restricted regression is zt= 1+ 4x4t+ut
In the F-test formula, T=144, k=4, m=2, RRSS=436.1, URSS=397.2
F-test statistic = 6.68. Critical value is an F(2,140) = 3.07 (5%) and 4.79 (1%).
Conclusion: Reject H0.
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