Real Estate Modelling and Forecasting By Chris Brooks_4 pdf

An overview of regression analysis 107
Atermin

x
∗2
t
can be cancelled from the numerator and denominator of (4A.29),
and, recalling that x
∗
t
= (x
t
−
¯
x ), this gives the variance of the slope coefficient as
var(
ˆ
β) =
s
2

(x
t
−
¯
x)
2
(4A.30)
so that the standard error can be obtained by taking the square root of (4A.30):
SE(
ˆ

β) = s




1

(x
t
−
¯
x)
2
(4A.31)
Turning now to the derivation of the intercept standard error, this is much more
difficult than that of the slope standard error. In fact, both are very much easier
using matrix algebra, as shown in the following chapter. This derivation is therefore
offered in summary form.
It is possible to express ˆα as a function of the true α and of the disturbances, u
t
:
ˆα = α +

u
t


x
2
t

− x
t

x
t


T

x
2
t
−


x
t

2

(4A.32)
Denoting all the elements in square brackets as g
t
, (4A.32) can be written
ˆα − α =

u
t
g
t

(4A.33)
From (4A.15), the intercept variance would be written
var( ˆα) = E


u
t
g
t

2
=

g
2
t
E

u
2
t

= s
2

g
2
t
(4A.34)
Writing (4A.34) out in full for g

2
t
and expanding the brackets,
var( ¯α) =
s
2

T


x
2
t

2
− 2

x
t


x
2
t


x
t
+



x
2
t


x
t

2


T

x
2
t
−


x
t

2

2
(4A.35)
This looks rather complex, but, fortunately, if we take

x

2
t
outside the square
brackets in the numerator, the remaining numerator cancels with a term in the
denominator to leave the required result:
SE(ˆα) = s






x
2
t
T

(x
t
−
¯
x)
2
(4A.36)
5
Further issues in regression analysis
Learning outcomes
In this chapter, you will learn how to
●
construct models with more than one explanatory variable;

●
derive the OLS parameter and standard error estimators in the
multiple regression context;
●
determine how well the model fits the data;
●
understand the principles of nested and non-nested models;
●
test multiple hypotheses using an F -test;
●
form restricted regressions; and
●
test for omitted and redundant variables.
5.1 Generalising the simple model to multiple linear regression
Previously, a model of the following form has been used:
y
t
= α + βx
t
+ u
t
t = 1, 2, ,T (5.1)
Equation (5.1) is a simple bivariate regression model. That is, changes in
the dependent variable are explained by reference to changes in one single
explanatory variable x. What if the real estate theory or the idea that is
sought to be tested suggests that the dependent variable is influenced by
more than one independent variable, however? For example, simple esti-
mation and tests of the capital asset pricing model can be conducted using
an equation of the form of (5.1), but arbitrage pricing theory does not pre-
suppose that there is only a single factor affecting stock returns. So, to give

one illustration, REIT excess returns might be purported to depend on their
sensitivity to unexpected changes in
108
Further issues in regression analysis 109
(1) inflation;
(2) the differences in returns on short- and long-dated bonds;
(3) the dividend yield; or
(4) default risks.
Having just one independent variable would be no good in this case. It
would, of course, be possible to use each of the four proposed explanatory
factors in separate regressions. It is of greater interest, though, and it is also
more valid, to have more than one explanatory variable in the regression
equation at the same time, and therefore to examine the effect of all the
explanatory variables together on the explained variable.
It is very easy to generalise the simple model to one with k regressors
(independent variables). Equation (5.1) becomes
y
t
= β
1
+ β
2
x
2t
+ β
3
x
3t
+···+β
k

x
kt
+ u
t
,t= 1, 2, ,T (5.2)
The variables x
2t
, x
3t
, ,x
kt
are therefore a set of k − 1 explanatory variables
that are thought to influence y, and the coefficient estimates β
2
, β
3
, , β
k
are the parameters that quantify the effect of each of these explanatory
variables on y. The coefficient interpretations are slightly altered in the
multiple regression context. Each coefficient is now known as a partial
regression coefficient, interpreted as representing the partial effect of the
given explanatory variable on the explained variable, after holding con-
stant, or eliminating the effect of, all the other explanatory variables. For
example,
ˆ
β
2
measures the effect of x
2

on y after eliminating the effects of x
3
,
x
4
, ,x
k
. Stating this in other words, each coefficient measures the average
change in the dependent variable per unit change in a given independent
variable, holding all other independent variables constant at their average
values.
5.2 The constant term
In (5.2) above, astute readers will have noticed that the explanatory variables
are numbered x
2
, x
3
, . . . – i.e. the list starts with x
2
and not x
1
.So,whereis
x
1
? In fact, it is the constant term, usually represented by a column of ones
of length T :
x
1
=
⎡

⎢
⎢
⎢
⎣
1
1
.
.
.
1
⎤
⎥
⎥
⎥
⎦
(5.3)
110 Real Estate Modelling and Forecasting
Thus there is a variable implicitly hiding next to β
1
, which is a column
vector of ones, the length of which is the number of observations in the
sample. The x
1
in the regression equation is not usually written, in the same
way that one unit of p and two units of q would be written as ‘p + 2q’ and
not ‘1p + 2q’. β
1
is the coefficient attached to the constant term (which was
called α in the previous chapter). This coefficient can still be referred to as
the intercept, which can be interpreted as the average value that y would

take if all the explanatory variables took a value of zero.
A tighter definition of k, the number of explanatory variables, is prob-
ably now necessary. Throughout this book, k is defined as the number of
‘explanatory variables’ or ‘regressors’, including the constant term. This is
equivalent to the number of parameters that are estimated in the regression
equation. Strictly speaking, it is not sensible to call the constant an explana-
tory variable, since it does not explain anything and it always takes the same
values. This definition of k will be employed for notational convenience,
however.
Equation (5.2) can be expressed even more compactly by writing it in
matrix form:
y = Xβ + u (5.4)
where: y is of dimension T × 1;
X is of dimension T × k;
β is of dimension k × 1; and
u is of dimension T × 1.
The difference between (5.2) and (5.4) is that all the time observations have
been stacked up in a vector, and also that all the different explanatory
variables have been squashed together so that there is a column for each
in the X matrix. Such a notation may seem unnecessarily complex, but,
in fact, the matrix notation is usually more compact and convenient. So,
for example, if k is two – i.e. there are two regressors, one of which is the
constant term (equivalent to a simple bivariate regression y
t
= α + βx
t
+ u
t
)
– it is possible to write

⎡
⎢
⎢
⎢
⎣
y
1
y
2
.
.
.
y
T
⎤
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎣
1 x
21
1 x
22
.

.
.
.
.
.
1 x
2T
⎤
⎥
⎥
⎥
⎦

β
1
β
2

+
⎡
⎢
⎢
⎢
⎣
u
1
u
2
.
.

.
u
T
⎤
⎥
⎥
⎥
⎦
(5.5)
T × 1 T × 22×1 T × 1
so that the x
ij
element of the matrix X represents the j th time observa-
tion on the ith variable. Notice that the matrices written in this way are
Further issues in regression analysis 111
conformable – in other words, there is a valid matrix multiplication and
addition on the RHS.
1
5.3 How are the parameters (the elements of the β vector)
calculated in the generalised case?
Previously, the residual sum of squares,

ˆ
u
2
i
, was minimised with respect
to α and β. In the multiple regression context, in order to obtain estimates
of the parameters, β
1

, β
2
, ,β
k
, the RSS would be minimised with respect
to all the elements of β. Now, the residuals can be stacked in a vector:
ˆ
u =
⎡
⎢
⎢
⎢
⎣
ˆ
u
1
ˆ
u
2
.
.
.
ˆ
u
T
⎤
⎥
⎥
⎥
⎦

(5.6)
The RSS is still the relevant loss function, and would be given in a matrix
notation by equation (5.7):
L =
ˆ
u

ˆ
u = [
ˆ
u
1
ˆ
u
2
···
ˆ
u
T
]
⎡
⎢
⎢
⎢
⎣
ˆ
u
1
ˆ
u

2
.
.
.
ˆ
u
T
⎤
⎥
⎥
⎥
⎦
=
ˆ
u
2
1
+
ˆ
u
2
2
+···+
ˆ
u
2
T
=

ˆ

u
2
t
(5.7)
Using a similar procedure to that employed in the bivariate regression case –
i.e. substituting into (5.7), and denoting the vector of estimated parameters
as
ˆ
β – it can be shown (see the appendix to this chapter) that the coefficient
estimates will be given by the elements of the expression
ˆ
β =
⎡
⎢
⎢
⎣
ˆ
β
1
ˆ
β
2
.
.
.
ˆ
β
k
⎤
⎥

⎥
⎦
= (X

X)
−1
X

y (5.8)
If one were to check the dimensions of the RHS of (5.8), it would be observed
to be k × 1. This is as required, since there are k parameters to be estimated
by the formula for
ˆ
β.
1
The above presentation is the standard way to express matrices in the time series
econometrics literature, although the ordering of the indices is different from that used in
the mathematics of matrix algebra (as presented in chapter 2 of this book). In the latter
case, x
ij
would represent the element in row i and column j, although, in the notation
used from this point of the book onwards, it is the other way around.
112 Real Estate Modelling and Forecasting
How are the standard errors of the coefficient estimates calculated,
though? Previously, to estimate the variance of the errors, σ
2
,anestima-
tor denoted by s
2
was used:

s
2
=

ˆ
u
2
t
T − 2
(5.9)
The denominator of (5.9) is given by T − 2, which is the number of degrees of
freedom for the bivariate regression model – i.e. the number of observations
minus two. This applies, essentially, because two observations are effectively
‘lost’ in estimating the two model parameters – i.e. in deriving estimates for
α and β. In the case in which there is more than one explanatory variable
plus a constant, and using the matrix notation, (5.9) would be modified to
s
2
=
ˆ
u

ˆ
u
T − k
(5.10)
where k = the number of regressors including a constant. In this case, k
observations are ‘lost’ as k parameters are estimated, leaving T − k degrees
of freedom. It can also be shown (see the appendix to this chapter) that the
parameter variance–covariance matrix is given by

var(
ˆ
β) = s
2
(X

X)
−1
(5.11)
The leading diagonal terms give the coefficient variances while the
off-diagonal terms give the covariances between the parameter estimates,
so that the variance of
ˆ
β
1
is the first diagonal element, the variance of
ˆ
β
2
is the second element on the leading diagonal and the variance of
ˆ
β
k
is
the kth diagonal element. The coefficient standard errors are simply given
therefore by taking the square roots of each of the terms on the leading
diagonal.
Example 5.1
The following model with three regressors (including the constant) is esti-
mated over fifteen observations,

y = β
1
+ β
2
x
2
+ β
3
x
3
+ u (5.12)
and the following data have been calculated from the original xs:
(X

X)
−1
=
⎡
⎢
⎣
2.03.5 −1.0
3.51.06.5
−1.06.54.3
⎤
⎥
⎦
, (X

y) =
⎡

⎢
⎣
−3.0
2.2
0.6
⎤
⎥
⎦
,
ˆ
u

ˆ
u = 10.96
Further issues in regression analysis 113
Calculate the coefficient estimates and their standard errors.
ˆ
β =
⎡
⎢
⎢
⎢
⎣
ˆ
β
1
ˆ
β
2
.

.
.
ˆ
β
k
⎤
⎥
⎥
⎥
⎦
= (X

X)
−1
X

y =
⎡
⎣
2.03.5 −1.0
3.51.06.5
−1.06.54.3
⎤
⎦
×
⎡
⎣
−3.0
2.2
0.6

⎤
⎦
=
⎡
⎣
1.10
−4.40
19.88
⎤
⎦
(5.13)
To calculate the standard errors, an estimate of σ
2
is required:
s
2
=
RSS
T − k
=
10.96
15 − 3
= 0.91 (5.14)
The variance–covariance matrix of
ˆ
β is given by
s
2
(X


X)
−1
= 0.91(X

X)
−1
=
⎡
⎣
1.82 3.19 −0.91
3.19 0.91 5.92
−0.91 5.92 3.91
⎤
⎦
(5.15)
The coefficient variances are on the diagonals, and the standard errors
are found by taking the square roots of each of the coefficient variances.
var(
ˆ
β
1
) = 1.82 SE(
ˆ
β
1
) = 1.35 (5.16)
var(
ˆ
β
2

) = 0.91 ⇔ SE(
ˆ
β
2
) = 0.95 (5.17)
var(
ˆ
β
3
) = 3.91 SE(
ˆ
β
3
) = 1.98 (5.18)
The estimated equation would be written
ˆ
y = 1.10 − 4.40x
2
+ 19.88x
3
(1.35) (0.95) (1.98)
(5.19)
In practice, fortunately, all econometrics software packages will estimate
the coefficient values and their standard errors. Clearly, though, it is still
useful to understand where these estimates came from.
5.4 A special type of hypothesis test: the t-ratio
Recall from equation (4.29) in the previous chapter that the formula under a
test of significance approach to hypothesis testing using a t-test for variable
i is
test statistic =

ˆ
β
i
− β
∗
i
SE

ˆ
β
i

(5.20)
114 Real Estate Modelling and Forecasting
If the test is
H
0
: β
i
= 0
H
1
: β
i
= 0
i.e. a test that the population parameter is zero against a two-sided alterna-
tive – this is known as a t-ratio test. Since β
∗
i
= 0, the expression in (5.20)

collapses to
test statistic =
ˆ
β
i
SE(
ˆ
β
i
)
(5.21)
Thus the ratio of the coefficient to its standard error, given by this expres-
sion, is known as the t-ratio or t-statistic. In the last example above, the
t-ratios associated with each of the three coefficients would be given by
ˆ
β
1
ˆ
β
2
ˆ
β
3
Coefficient 1.10 −4.40 19.88
SE 1.35 0.95 1.98
t-ratio 0.81 −4.63 10.04
Note that, if a coefficient is negative, its t-ratio will also be negative. In
order to test (separately) the null hypotheses that β
1
= 0, β

2
= 0 and β
3
= 0,
the test statistics would be compared with the appropriate critical value
from a t-distribution. In this case, the number of degrees of freedom, given
by T −k,isequalto15 − 3 = 12. The 5 per cent critical value for this two-
sided test (remember, 2.5 per cent in each tail for a 5 per cent test) is 2.179,
while the 1 per cent two-sided critical value (0.5 per cent in each tail) is 3.055.
Given these t-ratios and critical values, would the following null hypotheses
be rejected?
H
0
: β
1
= 0? No.
H
0
: β
2
= 0? Yes.
H
0
: β
3
= 0? Yes.
If H
0
is rejected, it would be said that the test statistic is significant. If
the variable is not ‘significant’ it means that, while the estimated value

of the coefficient is not exactly zero (e.g. 1.10 in the example above), the
coefficient is indistinguishable statistically from zero. If a zero was placed
in the fitted equation instead of the estimated value, this would mean
that, whatever happened to the value of that explanatory variable, the
dependent variable would be unaffected. This would then be taken to mean
that the variable is not helping to explain variations in y, and that it could
therefore be removed from the regression equation. For example, if the t-
ratio associated with x
3
had been 1.04 rather than 10.04, the variable would
Further issues in regression analysis 115
be classed as insignificant – i.e. not statistically different from zero). The
only insignificant term in the above regression is the intercept. There are
good statistical reasons for always retaining the constant, even if it is not
significant; see chapter 6.
It is worth noting that, for degrees of freedom greater than around twenty-
five, the 5 per cent two-sided critical value is approximately ±2. So, as a rule
of thumb (i.e. a rough guide), the null hypothesis would be rejected if the
t-statistic exceeds two in absolute value.
Some authors place the t-ratios in parentheses below the corresponding
coefficient estimates rather than the standard errors. Accordingly, one needs
to check which convention is being used in each particular application, and
also to state this clearly when presenting estimation results.
5.5 Goodness of fit statistics
5.5.1 R
2
It is desirable to have some measure of how well the regression model
actually fits the data. In other words, it is desirable to have an answer to the
question ‘How well does the model containing the explanatory variables
that was proposed actually explain variations in the dependent variable?’.

Quantities known as goodness of fit statistics are available to test how well
the sample regression function (SRF) fits the data – that is, how ‘close’ the
fitted regression line is to all the data points taken together. Note that
it is not possible to say how well the sample regression function fits the
population regression function – i.e. how the estimated model compares
with the true relationship between the variables – as the latter is never
known.
What measures might therefore make plausible candidates to be goodness
of fit statistics? A first response to this might be to look at the residual sum
of squares. Recall that OLS selected the coefficient estimates that minimised
this quantity, so the lower the minimised value of the RSS was, the better the
model fitted the data. Consideration of the RSS is certainly one possibility,
but the RSS is unbounded from above (strictly, it is bounded from above by
the total sum of squares – see below) – i.e. it can take any (non-negative)
value. So, for example, if the value of the RSS under OLS estimation was
136.4, what does this actually mean? It would be very difficult, by looking at
this number alone, to tell whether the regression line fitted the data closely
or not. The value of the RSS depends to a great extent on the scale of the
dependent variable. Thus one way to reduce the RSS pointlessly would be to
divide all the observations on y by ten!
116 Real Estate Modelling and Forecasting
In fact, a scaled version of the residual sum of squares is usually employed.
The most common goodness of fit statistic is known as R
2
. One way to define
R
2
is to say that it is the square of the correlation coefficient between y and
ˆ
y – that is, the square of the correlation between the values of the dependent

variable and the corresponding fitted values from the model. A correlation
coefficient must lie between −1 and +1 by definition. Since R
2
(defined in
this way) is the square of a correlation coefficient, it must lie between zero
and one. If this correlation is high, the model fits the data well, while, if the
correlation is low (close to zero), the model is not providing a good fit to the
data.
Another definition of R
2
requires a consideration of what the model is
attempting to explain. What the model is trying to do in effect is to explain
variability of y about its mean value,
¯
y. This quantity,
¯
y, which is more
specifically known as the unconditional mean of y, acts like a benchmark,
since, if the researcher had no model for y, he/she could do no worse than
to regress y on a constant only. In fact, the coefficient estimate for this
regression would be the mean of y. So, from the regression
y
t
= β
1
+ u
t
(5.22)
the coefficient estimate,
ˆ

β
1
,willbethemeanofy –i.e.
¯
y. The total variation
across all observations of the dependent variable about its mean value is
known as the total sum of squares, TSS, which is given by
TSS =

t
(y
t
−
¯
y)
2
(5.23)
The TSS can be split into two parts: the part that has been explained by
the model (known as the explained sum of squares, ESS) and the part that
the model was not able to explain (the RSS). That is,
TSS = ESS + RSS (5.24)

t
(y
t
−
¯
y)
2
=


t
(
ˆ
y
t
−
¯
y)
2
+

t
ˆ
u
2
t
(5.25)
Recall that the residual sum of squares can also be expressed as

t
(y
t
−
ˆ
y
t
)
2
since a residual for observation t is defined as the difference between the

actual and fitted values for that observation. The goodness of fit statistic
is given by the ratio of the explained sum of squares to the total sum of
squares,
R
2
=
ESS
TSS
(5.26)
Further issues in regression analysis 117
y
–
y
t
x
t
Figure 5.1
R
2
= 0
demonstrated by a
flat estimated line
but, since TSS = ESS + RSS,itisalsopossibletowrite
R
2
=
ESS
TSS
=
TSS − RSS

TSS
= 1 −
RSS
TSS
(5.27)
R
2
must always lie between zero and one (provided that there is a constant
term in the regression). This is intuitive from the correlation interpreta-
tion of R
2
given above, but, for another explanation, consider two extreme
cases:
RSS = TSS i.e. ESS = 0soR
2
= ESS/TSS = 0
ESS = TSS i.e. RSS = 0soR
2
= ESS/TSS = 1
In the first case, the model has not succeeded in explaining any of the
variability of y about its mean value, and hence the residual and total sums
of squares are equal. This would happen only when the estimated values
of all the coefficients were exactly zero. In the second case, the model has
explained all the variability of y about its mean value, which implies that
the residual sum of squares will be zero. This would happen only in the case
in which all the observation points lie exactly on the fitted line. Neither of
these two extremes is likely in practice, of course, but they do show that
R
2
is bounded to lie between zero and one, with a higher R

2
implying,
everything else being equal, that the model fits the data better.
To sum up, a simple way (but crude, as explained next) to tell whether
the regression line fits the data well is to look at the value of R
2
.Avalueof
R
2
close to one indicates that the model explains nearly all the variability
of the dependent variable about its mean value, while a value close to zero
indicates that the model fits the data poorly. The two extreme cases, in
which R
2
= 0 and R
2
= 1, are indicated in figures 5.1 and 5.2 in the context
of a simple bivariate regression.
118 Real Estate Modelling and Forecasting
y
t
x
t
Figure 5.2
R
2
= 1 when all data
points lie exactly on
the estimated line
Example 5.2 Measuring goodness of fit

We now estimate the R
2
for equation (4.28) applying formula (5.27). RSS =
1214.20, TSS = 2550.59.
R
2
= 1 −
RSS
TSS
= 1 −
1214.20
2550.59
= 0.52
Equation (4.28) explains 52 per cent of the variability of rent growth. For a
bivariate regression model, this would usually be considered a satisfactory
performance.
5.5.2 Problems with R
2
as a goodness of fit measure
R
2
is simple to calculate and intuitive to understand, and provides a broad
indication of the fit of the model to the data. There are a number of prob-
lems with R
2
as a goodness of fit measure, however, which are outlined in
box 5.1.
Box 5.1 Disadvantages of R
2
(1) R

2
is defined in terms of variation about the mean of y, so, that if a model is
reparameterised (rearranged) and the dependent variable changes, R
2
will
change, even if the second model is a simple rearrangement of the first, with
identical RSS. Thus it is not sensible to compare the value of R
2
between models
with different dependent variables.
(2) R
2
never falls if more regressors are added to the regression. For example,
consider the following two models:
regression 1: y = β
1
+ β
2
x
2
+ β
3
x
3
+ u (5.28)
regression 2: y = β
1
+ β
2
x

2
+ β
3
x
3
+ β
4
x
4
+ u (5.29)
Further issues in regression analysis 119
R
2
will always be at least as high for regression 2 relative to regression 1. The R
2
from regression 2 would be exactly the same as that for regression 1 only if the
estimated value of the coefficient on the new variable were exactly zero – i.e.
ˆ
β
4
= 0.Inpractice,
ˆ
β
4
will always be non-zero, even if not significantly so, and
thus in practice R
2
always rises as more variables are added to a model. This
feature of R
2

essentially makes it impossible to use as a determinant of whether
a given variable should be present in the model or not.
(3) R
2
quite often takes on values of 0.9 or higher for time series regressions, and
hence it is not good at discriminating between models, as a wide array of models
will frequently have broadly similar (and high) values of R
2
.
5.5.3 Adjusted R
2
In order to get round the second of these three problems, a modification
to R
2
is often made that takes into account the loss of degrees of freedom
associated with adding extra variables. This is known as
¯
R
2
, or adjusted R
2
,
which is defined as
¯
R
2
= 1 −

T − 1
T − k

(1 − R
2
)

(5.30)
where k is the number of parameters to be estimated in the model and T
is the sample size. If an extra regressor (variable) is added to the model, k
increases and, unless R
2
increases by a more than offsetting amount,
¯
R
2
will
actually fall. Hence
¯
R
2
can be used as a decision-making tool for determining
whether a given variable should be included in a regression model or not,
with the rule being: include the variable if
¯
R
2
rises and do not include it if
¯
R
2
falls.
There are still problems with the maximisation of

¯
R
2
, however, as a
criterion for model selection.
(1) It is a ‘soft’ rule, implying that, by following it, the researcher will
typically end up with a large model, containing a lot of marginally
significant or insignificant variables.
(2) There is no distribution available for
¯
R
2
or R
2
, so hypothesis tests cannot
be conducted using them. The implication is that one can never tell
whether the R
2
or the
¯
R
2
from one model is significantly higher than
that of another model in a statistical sense.
5.6 Tests of non-nested hypotheses
All the hypothesis tests conducted thus far in this book have been in the
context of ‘nested’ models. This means that, in each case, the test involved
120 Real Estate Modelling and Forecasting
imposing restrictions on the original model to arrive at a restricted formu-
lation that would be a subset of, or nested within, the original specification.

Sometimes, however, it is of interest to compare between non-nested mod-
els. For example, suppose that there are two researchers working indepen-
dently, each with a separate real estate theory for explaining the variation
in some variable, y
t
. The respective models selected by the researchers could
be
y
t
= α
1
+ α
2
x
2t
+ u
t
(5.31)
y
t
= β
1
+ β
2
x
3t
+ v
t
(5.32)
where u

t
and v
t
are iid error terms. Model (5.31) includes variable x
2
but not
x
3
, while model (5.32) includes x
3
but not x
2
. In this case, neither model can
be viewed as a restriction of the other, so how then can the two models be
compared as to which better represents the data, y
t
? Given the discussion in
the previous section, an obvious answer would be to compare the values of
R
2
or adjusted R
2
between the models. Either would be equally applicable
in this case, since the two specifications have the same number of RHS
variables. Adjusted R
2
could be used even in cases in which the number
of variables was different in the two models, since it employs a penalty
term that makes an allowance for the number of explanatory variables.
Adjusted R

2
is based upon a particular penalty function, however (that is,
T −k appears in a specific way in the formula). This form of penalty term
may not necessarily be optimal.
Moreover, given the statement above that adjusted R
2
is a soft rule, it is
likely on balance that use of it to choose between models will imply that
models with more explanatory variables are favoured. Several other similar
rules are available, each having more or less strict penalty terms; these are
collectively known as ‘information criteria’. These are explained in some
detail in chapter 8, but suffice to say for now that a different strictness of
the penalty term will in many cases lead to a different preferred model.
An alternative approach to comparing between non-nested models would
be to estimate an encompassing or hybrid model. In the case of (5.31) and
(5.32), the relevant encompassing model would be
y
t
= γ
1
+ γ
2
x
2t
+ γ
3
x
3t
+ w
t

(5.33)
where w
t
is an error term. Formulation (5.33) contains both (5.31) and
(5.32) as special cases when γ
3
and γ
2
are zero, respectively. Therefore a
test for the best model would be conducted via an examination of the sig-
nificances of γ
2
and γ
3
in model (5.33). There will be four possible outcomes
(box 5.2).
Further issues in regression analysis 121
Box 5.2 Selecting between models
(1) γ
2
is statistically significant but γ
3
is not. In this case, (5.33) collapses to (5.31),
and the latter is the preferred model.
(2) γ
3
is statistically significant but γ
2
is not. In this case, (5.33) collapses to (5.32),
and the latter is the preferred model.

(3) γ
2
and γ
3
are both statistically significant. This would imply that both x
2
and x
3
have incremental explanatory power for y, in which case both variables should be
retained. Models (5.31) and (5.32) are both ditched and (5.33) is the preferred
model.
(4) Neither γ
2
nor γ
3
is statistically significant. In this case, none of the models can be
dropped, and some other method for choosing between them must be employed.
There are several limitations to the use of encompassing regressions to
select between non-nested models, however. Most importantly, even if mod-
els (5.31) and (5.32) have a strong theoretical basis for including the RHS
variables that they do, the hybrid model may be meaningless. For example,
it could be the case that real estate theory suggests that y could either follow
model (5.31) or model (5.32), but model (5.33) is implausible.
In adition, if the competing explanatory variables x
2
and x
3
are highly
related – i.e. they are near-collinear – it could be the case that, if they are
both included, neither γ

2
nor γ
3
is statistically significant, while each is
significant in its separate regressions (5.31) and (5.32); see chapter 6 for an
explanation of why this may happen.
An alternative approach is via the J -encompassing test due to Davidson
and MacKinnon (1981). Interested readers are referred to their work or to
Gujarati (2009) for further details.
Example 5.3 A multiple regression in real estate
Amy, Ming and Yuan (2000) study the Singapore office market and focus
on obtaining empirical estimates for the natural vacancy rate and rents
utilising existing theoretical frameworks. Their empirical analysis includes
the estimation of different specifications for rents. For their investigation,
quarterly data are available. One of the models they estimate is given by
equation (5.34),
%R
t
= β
0
+ β
1
%E
t
− β
2
V
t−1
(5.34)
where % denotes a percentage change (over the previous quarter), R

t
is the
nominal rent (hence %R
t
is the percentage change in nominal rent this
quarter over the preceding one), E
t
is the operating costs (due to data limita-
tions, the authors approximate this variable with the consumer price index;
the CPI reflects the cost-push elements in an inflationary environment as
122 Real Estate Modelling and Forecasting
landlords push for higher rents to cover inflation and expenses) and V
t−1
is
the vacancy rate (in per cent) in the previous quarter. The fitted model is
%
ˆ
R
t
= 6.21 + 2.07(%E
t
) − 0.54V
t−1
(5.35)
(2.7) (2.5) (−3.0)
Adjusted
¯
R
2
= 0.23

According to the above results, if the vacancy rate in the previous quarter
fell by 1 per cent, the rate of nominal growth will increase by 0.54 per
cent. This is considered a rather small sensitivity. An increase in the CPI of
1 per cent will push up the rate of nominal rent growth by 2.07 per cent.
The t-statistics in parentheses confirm that the parameters are statistically
significant.
The above model explains approximately 23 per cent of the variation in
nominal rent growth, which means that model (5.35) has quite low explana-
tory power. Both the low explanatory power and the small sensitivity of
rents to vacancy are perhaps a result of model misspecification, which the
authors detect and attempt to address in their paper. We consider such
issues of model misspecification in the following chapter.
An alternative model that Amy, Ming and Yuan run is
%RR
t
= β
0
+ β
2
V
t
+ u
t
(5.36)
This is a bivariate regression model; %RR
t
is the quarterly percentage
change in real rents (note that, in equation (5.34), nominal growth was
used). The following equation is the outcome:
%RR

t
= 18.53 − 1.50V
t
(5.37)
(1.67) (−3.3)
Adjusted R
2
= 0.21
In equation (5.37), the vacancy takes the expected negative sign and the
coefficient suggests that a 1 per cent rise in vacancy will, on average, reduce
the rate of growth of real rents by 1.5 per cent. The sensitivity of rent growth
to vacancy is greater than that in the previous model. The explanatory power
remains low, however.
Although we have not completed the treatment of regression analysis, one
may ask whether we can take a view as to which model is more appropriate
to study office rents in Singapore.
This book equips the reader with the tools to answer this question, in
particular by means of the tests we discuss in the next chapter and the
evaluation of forecast performance in later chapters. On the basis of the
Further issues in regression analysis 123
information we have for these two models, however, some observations can
be made.
(1) We would prefer the variables to be in real terms (adjusted for inflation),
as for the rent series in equation (5.36).
(2) The models seem to have similar explanatory power but equation (5.34)
has two drivers. Caution should be exercised, however. We said earlier
that the adjusted R
2
can be used for comparisons only if the dependent
variable is the same (which is not the case here, since the dependent

variables differ). In this case, a comparison can tentatively be made,
given that the dependent variables are not entirely different (which
would have been the case if we had been modelling the percentage
change in rents and the level of rents, for example).
Going back to our earlier point that more testing is required, the authors
report misspecification problems in their paper, and hence none of the
models scores particularly well. Based on this information, we would choose
equation (5.36), because of point (1) above and the low adjusted R
2
of the
multiple regression model (5.35).
5.7 Data mining and the true size of the test
Recall that the probability of rejecting a correct null hypothesis is equal
to the size of the test, denoted α. The possibility of rejecting a correct null
hypothesis arises from the fact that test statistics are assumed to follow
a random distribution and hence take on extreme values that fall in the
rejection region some of the time by chance alone. A consequence of this
is that it will almost always be possible to find significant relationships
between variables if enough variables are examined. For example, suppose
that a dependent variable y
t
and twenty explanatory variables x
2t
, , x
21t
(excluding a constant term) are generated separately as independent nor-
mally distributed random variables. Then y is regressed separately on each
of the twenty explanatory variables plus a constant, and the significance
of each explanatory variable in the regressions is examined. If this experi-
ment is repeated many times, on average one of the twenty regressions will

have a slope coefficient that is significant at the 5 per cent level for each
experiment. The implication is that, if enough explanatory variables are
employed in a regression, often one or more will be significant by chance
alone. More concretely, it could be stated that, if an α per cent size of test
is used, on average one in every (100/α) regressions will have a significant
slope coefficient by chance alone.
124 Real Estate Modelling and Forecasting
Trying many variables in a regression without basing the selection of the
candidate variables on a real estate or economic theory is known as ‘data
mining’ or ‘data snooping’. The result in such cases is that the true sig-
nificance level will be considerably greater than the nominal significance
level assumed. For example, suppose that twenty separate regressions are
conducted, of which three contain a significant regressor, and a 5 per cent
nominal significance level is assumed; then the true significance level would
be much higher (e.g. 25 per cent). Therefore, if the researcher then shows
only the results for the regression containing the final three equations and
states that they are significant at the 5 per cent level, inappropriate conclu-
sions concerning the significance of the variables would result.
As well as ensuring that the selection of candidate regressors for inclusion
in a model is made on the basis of real estate theory, another way to avoid
data mining is by examining the forecast performance of the model in an
‘out-of-sample’ data set (see chapters 8 and 9). The idea, essentially, is that
a proportion of the data is not used in model estimation but is retained
for model testing. A relationship observed in the estimation period that is
purely the result of data mining, and is therefore spurious, is very unlikely
to be repeated for the out-of-sample period. Therefore models that are the
product of data mining are likely to fit very poorly and to give very inaccu-
rate forecasts for the out-of-sample period. This topic will be elaborated in
subsequent chapters.
5.8 Testing multiple hypotheses: the F-test

The t-test was used to test single hypotheses – i.e. hypotheses involving
only one coefficient. What if it is of interest to test more than one coeffi-
cient simultaneously, however? For example, what if a researcher wanted
to determine whether a restriction that the coefficient values for β
2
and β
3
are both unity could be imposed, so that an increase in either one of the two
variables x
2
or x
3
would cause y to rise by one unit? The t-testing framework
is not sufficiently general to cope with this sort of hypothesis test. Instead, a
more general framework is employed, centring on an F -test. Under the
F -test framework, two regressions are required, known as the unrestricted
and the restricted regressions. The unrestricted regression is the one in which
the coefficients are freely determined by the data, as has been constructed
previously. The restricted regression is the one in which the coefficients are
restricted – i.e. the restrictions are imposed on some βs. Thus the F -test
approach to hypothesis testing is also termed restricted least squares, for
obvious reasons.
Further issues in regression analysis 125
The residual sums of squares from each regression are determined, and
the two residual sums of squares are ‘compared’ in the test statistic. The F -
test statistic for testing multiple hypotheses about the coefficient estimates
is given by
test statistic =
RRSS − URSS
URSS

×
T − k
m
(5.38)
where the following notation applies:
URSS = residual sum of squares from unrestricted regression;
RRSS = residual sum of squares from restricted regression;
m = number of restrictions;
T = number of observations; and
k = number of regressors in unrestricted regression,
including a constant.
The most important part of the test statistic to understand is the numer-
ator expression, RRSS − URSS. To see why the test centres around a compar-
ison of the residual sums of squares from the restricted and unrestricted
regressions, recall that OLS estimation involves choosing the model that
minimises the residual sum of squares, with no constraints imposed. If,
after imposing constraints on the model, a residual sum of squares results
that is not much higher than the unconstrained model’s residual sum of
squares, it would be concluded that the restrictions were supported by the
data. On the other hand, if the residual sum of squares increased consid-
erably after the restrictions were imposed, it would be concluded that the
restrictions were not supported by the data and therefore that the hypoth-
esis should be rejected.
It can be further stated that RRSS ≥ URSS. Only under a particular set
of very extreme circumstances will the residual sums of squares for the
restricted and unrestricted models be exactly equal. This would be the case
when the restriction was already present in the data, so that it is not really a
restriction at all (it would be said that the restriction is ‘not binding’ – i.e. it
does not make any difference to the parameter estimates). So, for example,
if the null hypothesis is H

0
: β
2
= 1 and β
3
= 1, then RRSS = URSS only in the
case in which the coefficient estimates for the unrestricted regression are
ˆ
β
2
= 1 and
ˆ
β
3
= 1. Of course, such an event is extremely unlikely to occur
in practice.
Example 5.4
In the previous chapter, we estimated a bivariate model of real rent growth
for UK offices (equation 4.10). The single explanatory variable was the growth
126 Real Estate Modelling and Forecasting
in employment and financial and business services. We now extend this
model to include GDP growth as another explanatory variable. There is an
argument in the existing literature suggesting that employment is not the
only factor that will affect rent growth but also an output measure that
better captures turnover and profitability.
2
The results of the multiple regression model of real office rent growth in
the United Kingdom are given in equation (5.39) with t-statistics in paren-
theses. In this example of modelling UK office rents, we have also extended
the sample by one more year, to 2006, compared with that in the previous

chapter. From the results in this equation (estimated for the sample period
1979 to 2006), GDPg makes an incremental contribution to explain growth
in real office rents:
R
ˆ
Rg
t
=−11.53 + 2.52EFBSg
t
+ 1.75GDPg
t
(5.39)
(−4.9) (3.7) (2.1)
We would like to test the hypothesis that the coefficients on both GDP
growth (GDPg
t
) and employment growth (EFBSg
t
) are zero. The unrestricted
and restricted equations are, respectively,
RRg
t
= α + β
1
EFBSg
t
+ β
2
GDPg
t

+ u
t
(5.40)
RRg
t
= α + u
t
(5.41)
The RSS values for the unrestricted and restricted equation are 1,078.26
and 2,897.73, respectively. The number of observations, T , is twenty-eight.
The number of restrictions, m, is two and the number of parameters to
be estimated in the unrestricted equation, k, is three. Applying the for-
mula (5.38), we get the value of 21.09 for the test statistic. The test statistic
will follow an F (m, T − k)orF (2, 25), with critical value 3.39 at the 5 per
cent significance level. The test statistic clearly exceeds the critical value at
5 per cent, and hence the null hypothesis is rejected. Therefore the coeffi-
cients are not jointly zero.
We would now like to test the hypothesis that the coefficients on EFBS
and GDP are equal and thus that the two variables have the same impact on
real rent growth – that is, β
1
= β
2
. The unrestricted and restricted equations
are, respectively,
RRg
t
= α + β
1
EFBSg

t
+ β
2
GDP
t
+ u
t
(5.42)
RRg
t
= α + β
1
(EFBSg
t
+ GDPg
t
) + u
t
(5.43)
2
The GDP data are taken from the Office for National Statistics.
Further issues in regression analysis 127
The RSS for the unrestricted and restricted equation are 1,078.26 and
1,092.81, respectively. The number of observations, T , is twenty-eight. The
number of restrictions, m, is two and the number of parameters to be esti-
mated in the unrestricted equation is three. Applying the formula (5.38), we
get the test statistic value of 0.46. The F (m, T − k) or F (1, 25) critical value
is 4.24 at the 5 per cent significance level. The test statistic is considerably
lower than the critical value, and hence the null hypothesis is not rejected.
Therefore the coefficients on EFBSg and GDPg (the slopes) are not statistically

significantly different from one another.
5.8.1 The relationship between the t- and the F-distributions
Any hypothesis that can be tested with a t-test could also have been tested
using an F -test, but not the other way around. Accordingly, single hypothe-
ses involving one coefficient can be tested using a t -oranF -test, but mul-
tiple hypotheses can be tested only using an F -test. For example, consider
the hypothesis
H
0
: β
2
= 0.5
H
1
: β
2
= 0.5
This hypothesis could have been tested using the usual t-test,
test stat =
ˆ
β
2
− 0.5
SE(
ˆ
β
2
)
(5.44)
or it could be tested in the framework above for the F -test. Note that the

two tests always give the same conclusion, since the t-distribution is just a
special case of the F -distribution, as demonstrated in box 5.3.
Box 5.3 The t- and F -distributions compared
●
Consider any random variable Z that follows a t-distribution with T − k degrees of
freedom, and square it. The square of the t is equivalent to a particular form of the
F -distribution:
Z
2
∼ t
2
(T − k) then also Z
2
∼ F (1,T −k)
●
Thus the square of a t-distributed random variable with T − k degrees of freedom
also follows an F -distribution with one and T − k degrees of freedom.
●
This relationship between the t- and the F -distributions will always hold.
●
The F -distribution has only positive values and is not symmetrical.
●
Therefore the null is rejected only if the test statistic exceeds the critical F -value,
although the test is a two-sided one in the sense that rejection will occur if
ˆ
β
2
is
significantly bigger or significantly smaller than 0.5.
128 Real Estate Modelling and Forecasting

5.8.2 Determining the number of restrictions, m
How is the appropriate value of m decided in each case? Informally, the
number of restrictions can be seen as ‘the number of equality signs under
the null hypothesis’. To give some examples:
H
0
: hypothesis number of restrictions,m
β
1
+ β
2
= 21
β
2
= 1 and β
3
=−12
β
2
= 0,β
3
= 0 and β
4
= 03
At first glance, you may have thought that, in the first of these cases, the
number of restrictions was two. In fact, there is only one restriction that
involves two coefficients. The number of restrictions in the second two
examples is obvious, as they involve two and three separate component
restrictions, respectively.
The last of these three examples is particularly important. If the model is

y = β
1
+ β
2
x
2
+ β
3
x
3
+ β
4
x
4
+ u (5.45)
then the null hypothesis of
H
0
: β
2
= 0 and β
3
= 0 and β
4
= 0
is tested by ‘the’ regression F -statistic. It tests the null hypothesis that all the
coefficients except the intercept coefficient are zero. This test is sometimes
called a test for ‘junk regressions’, since, if this null hypothesis cannot be
rejected, it would imply that none of the independent variables in the model
was able to explain variations in y.

Note the form of the alternative hypothesis for all tests when more than
one restriction is involved:
H
1
: β
2
= 0orβ
3
= 0orβ
4
= 0
In other words, ‘and’ occurs under the null hypothesis and ‘or’ under the
alternative, so that it takes only one part of a joint null hypothesis to be
wrong for the null hypothesis as a whole to be rejected.
5.8.3 Hypotheses that cannot be tested with either an F-orat-test
It is not possible to test hypotheses that are not linear or that are multiplica-
tive using this framework; for example, H
0
: β
2
β
3
= 2,orH
0
: β
2
2
= 1, cannot
be tested.
Further issues in regression analysis 129

5.9 Omission of an important variable
What would the effects be of excluding from the estimated regression a
variable that is a determinant of the dependent variable? For example,
suppose that the true, but unknown, data-generating process is represented
by
y
t
= β
1
+ β
2
x
2t
+ β
3
x
3t
+ β
4
x
4t
+ β
5
x
5t
+ u
t
(5.46)
but that the researcher estimates a model of the form
y

t
= β
1
+ β
2
x
2t
+ β
3
x
3t
+ β
4
x
4t
+ u
t
(5.47)
– i.e. with the variable x
5t
omitted from the model. The consequence would
be that the estimated coefficients on all the other variables will be biased
and inconsistent unless the excluded variable is uncorrelated with all the
included variables. Even if this condition is satisfied, the estimate of the
coefficient on the constant term will be biased, which would imply that any
forecasts made from the model would be biased. The standard errors will also
be biased (upwards), and hence hypothesis tests could yield inappropriate
inferences. Further intuition is offered by Dougherty (1992, pp. 168–73).
Example 5.5
Tests for omitted variables are very difficult to conduct in practice because

we will usually not be able to observe the omitted variables, or we may not
even be aware of their relevance. For illustration of what we would do if
we did have a candidate omitted variable, however, in the context of our
previous example of modelling real office rent growth, we now test whether
we have omitted output in the financial and business services sectors (OFBS).
3
One could argue that GDP is a more general measure of activity in those
business sectors that require office space whereas output in financial and
business services is more relevant to examine activity and occupier demand
in office-using industries. We run an F -test based on the sum of squared
residuals in the restricted and unrestricted equations.
Before presenting the results, a note should be made about the sample
size. For our reference equation (5.39), the sample period is 1979 to 2006,
which comprises twenty-eight observations. When we examine the signifi-
cance of the omitted OFBS series, the sample size shrinks, since official data
for OFBS start only in 1983 and, since we use growth rates, the sample com-
mences in 1984. Hence the estimations and tests below take place for the
3
The data are from the Office for National Statistics.
130 Real Estate Modelling and Forecasting
sample period 1984 to 2006, which leads to different parameter estimates.
Admittedly, this is a restricted sample period, but the objective here is more
to illustrate the application of the test.
Unrestricted equation (1984–2006):
ˆ
RRg
t
=−17.68 + 2.45EFBSg
t
+ 2.71GDPg

t
+ 0.79OFBSg
t
(5.48)
(6.35) (3.76) (1.92) (0.87)
where OFBSg is the annual percentage growth in OFBS,R
2
= 0.76, adj.
R
2
= 0.72, URSS = 619.46, k (number of regressors) = 4 and m (number of
added variables) = 1.
Restricted equation (1984–2006):
ˆ
RRg
t
=−17.06 + 2.53EFBSg
t
+ 3.58GDPg
t
(5.49)
(6.37) (3.96) (3.61)
R
2
= 0.75,adj.R
2
= 0.72, RRSS = 643.98, T = 23.
Before conducting the F -test, we make the following observations.
●
The value of the intercept parameter estimate has not changed much,

which provides an indication that the added variable does not bring
further information into the model.
●
The coefficient that is most affected is that of GDPg.HenceOFBSg very
probably conveys similar information to that of GDPg. Indeed, their cor-
relation is strong (the correlation coefficient is 0.78), whereas OFBSg is
more moderately correlated with EFBSg (correlation coefficient = 0.48).
Therefore GDPg and OFBSg are collinear, and this is why their t-ratios are
not significant; this issue is discussed in detail in the following chapter.
●
We observe no impact on the coefficients of determination, and in fact
the adjusted R
2
of the unrestricted model is lower – a signal that the
added variable does not contribute anything.
The F -test statistic is
643.98−619.46
619.46
×
23−4
1
= 0.75. The critical value for
F (1,19) at the 5 per cent level is 4.38. Since the computed value is less
than the critical value, we do not reject the null hypothesis that OFBSg does
not belong to the equation at the 5 per cent significance level.
5.10 Inclusion of an irrelevant variable
Suppose now that the researcher makes the opposite error to that in section
5.9 – i.e. the true DGP is represented by
y
t

= β
1
+ β
2
x
2t
+ β
3
x
3t
+ β
4
x
4t
+ u
t
(5.50)
Further issues in regression analysis 131
but the researcher estimates a model of the form
y
t
= β
1
+ β
2
x
2t
+ β
3
x

3t
+ β
4
x
4t
+ β
5
x
5t
+ u
t
(5.51)
thus incorporating the superfluous or irrelevant variable x
5t
.Asx
5t
is irrele-
vant, the expected value of β
5
is zero, although, in any practical application,
its estimated value is very unlikely to be exactly zero. The consequence of
including an irrelevant variable would be that the coefficient estimators
would still be consistent and unbiased, but the estimators would be inef-
ficient. This would imply that the standard errors for the coefficients are
likely to be inflated relative to the values that they would have taken if the
irrelevant variable had not been included. Variables that would otherwise
have been marginally significant may no longer be so in the presence of
irrelevant variables. In general, it can also be stated that the extent of the
loss of efficiency will depend positively on the absolute value of the corre-
lation between the included irrelevant variable and the other explanatory

variables.
Summarising the last two sections, it is evident that, when trying to deter-
mine whether to err on the side of including too many or too few variables
in a regression model, there is an implicit trade-off between inconsistency
and efficiency. Many researchers would argue that, while, in an ideal world,
the model will incorporate precisely the correct variables – no more and
no less – the former problem is more serious than the latter, and therefore,
in the real world, one should err on the side of incorporating marginally
significant variables.
Example 5.6
In our model for UK office rents, we test whether GDPg is irrelevant by
computing an F -test. The null hypothesis is that GDPg does not belong to
the equation (5.39) or that the coefficient on GDP
g
is zero. Similar to example
5.5 above, we run the unrestricted and restricted equations. The unrestricted
equation is equation (5.39). For that equation, the following statistics were
obtained: R
2
= 0.58, adj. R
2
= 0.55, RRSS = 1,078.26.
Restricted equation:
ˆ
RRg
t
=−9.54 + 3.25EFBSg
t
(5.52)
(1.80) (5.18)

R
2
= 0.51, adj. R
2
= 0.49, RRSS = 1,268.38. We observe that the adj. R
2
has not dropped much in the restricted equation and that the size of
the coefficient on EFBSg
t
has increased, as it now picks up more influ-
ences that were previously explained by GDPg
t
.TheF -test statistic is
1268.38−1078.26
1078.26
×
28−3
1
= 4.41.