THE DYNAMIC LINEAR REGRESSION MODEL

The linear regression model can be viewed as a statistical model derived by reduction from the joint distribution DZ, ..., Z7; w, where Z,=,.X;’, and {Z,,teT} is assumed to be a normal,

prediction) of the DLR model

The dependence in the sample raises the issue of introducing the concept

of dependent random variables or stochastic processes For this reason the reader is advised to refer back to Chapter 8 where the idea of a stochastic process and related concepts are discussed in some detail before proceeding further with the discussion which follows

The linear regression model can be viewed as a statistical model derived

by reduction from the joint distribution D(Z, ., Z7; w), where

Z,=(),.X;)’, and {Z,,teT} is assumed to be a normal, independent and

identically distributed (NIID) vector stochastic process For the purposes

of the present chapter we need to extend this to a more general stochastic process in order to take the dependence, which constitutes additional systematic information, into consideration

In Chapter 21 the identically distributed component was relaxed leading

to time varying parameters The main aim of the present chapter is to relax the independence component but retain the identically distributed assumption in the form of stationarity

In Section 23.1 the DLR is specified assuming that {Z,,teT} is a

stationary, asymptotically independent normal process Some of the arguments discussed in Chapter 22 in terms of the respecification approach will be considered more formally Section 23.2 considers the estimation of

the DLR using approximate MLE’s Sections 23.3 and 23.4 discuss

526

23.1 Specification 527 misspecification and specification testing in the context of the DLR model, respectively Section 23.5 considers the problem of prediction In Section 23.6 the empirical econometric model constructed in Section 23.4 is used to

‘explain’ the misspecification results derived in Chapters 19-22

23.1 Specification

In defining the linear regression model {Z,,t¢ 1} was assumed to be an

independent and identically distributed (IID) multivariate normal process

In defining the dynamic linear regression (DLR) model {Z,,teT} is assumed to be a stationary, asymptotically independent normal process

That is, the assumption of identically distributed has been extended to

that of stationarity and the independence assumption to that of asymptotic independence (see Chapter 8) In particular, {Z„,€ I} is assumed to have the following autoregressive representation:

Z,= > A()Z,_;¡+E,, t>m, (23.1) i=1

where

i=1 Z)_ Hư 1? Z,- 2s Z1);

_ di¡()— Ai¿ ) a5,(i) A:¿) and

characterised by realisations which, apart from the apparent existence of a local trend, the time path seems similar in the various parts of the realisation In such cases the differencing transformation

Al=(I—LHI, where ỨX,=X, „ i=1,2,

can be used to transform the original series to a stationary one (see Chapter 8)

The distribution underlying (1) is D(Z,/Z?_,:y) arising from the sequential decomposition of the joint distribution of (Z,, Z,, , Z7):

D(Z.,.Z; Z.+: ý)

=D(7, /Z„:Ú) || DƯ/⁄24.\ Z4 „Ú) (23.5)

t=m +]

D(Z,, , Zn W) refers to the joint distribution of the first m observation

interpreted as the initial conditions Asymptotic independence enables us to argue that asymptotically the effect of the initial conditions is negligible One important implication of this is that we can ignore the first m observations and treat r=m+1, , T as being the sample for statistical inference purposes In what follows this ‘solution’ will be adopted for expositional purposes because ‘proper’ treatment of the initial conditions can complicate the argument without adding to our understanding The statistical generating mechanism (GM) of the dynamic linear

regression model is based on the conditional distribution D(y,/X,, Z°_ 1: W,) related to D(Z,/Z°_,; p) via

D(Z,/Z)- 45 W= Dy /X 5 Zp 1s Wy D(X, /Z) 15 a) (23.6)

The systematic component is defined by

Hy = Ely,/a(¥P 1), XP =x?) Box + 2 (%)¥,-;+ BX), t>m,

(23.7) where

These two properties show that u, is a martingale difference process relative

to ¥ with bounded variance, ie an innovation process (see Section 8) Moreover, the non-systematic component is also orthogonal to the systematic component, i.e

The properties ET 1-ET3 can be verified directly using the properties of the conditional expectation discussed in Section 7.2 In view of the equality

we can deduce that

oe TO

for Up_ 4, =(uy 4 Uy 2 My)

i.e u, is not predictable from its own past

This property extends the notion of a white-noise process encountered so far (see Granger (1980))

As argued in Chapter 17, the parameters of interest are the parameters in terms of which the statistical GM is defined unless stated otherwise These parameters should be defined more precisely as statistical parameters of interest with the theoretical parameters of interest being functions of the former In the present context the statistical parameters of interest are 0* =

H(w,) where 0* =(Bo, ịm ⁄ị Sms FO):

The normality of D(Z,/Z°:; ý) implies that w, end w, are variation free and thus X, is weakly exogenous with respect to @ This suggests that 6* can

be estimated efficiently without any reference to the marginal distribution

D(X,/Z°_,: 2) The presence of Y?_, in this marginal distribution,

however, raises questions in the context of prediction because of the feedback from the lagged y,s In order to be able to treat the x,s as given when predicting \, we need to ensure that no such feedback exists For this purpose we need to assume that

D(X, /Zo_ 1: Wo) =D(X,/X°_ 43a) t=m4+1, ,T (23.14) i.e y, does not Granger cause X, (see Engle et al (1983) for a more detailed

discussion) Weak exogeneity of X, with respect to 6* when supplemented with Granger non-causality, as defined in (14), is called strong exogeneity Note that in the present context Granger non-causality is equivalent to

in (1), which suggests that the assumption is testable

In the case of the linear regression model it was argued that, although the joint distribution D(Z,; yw) was used to motivate the statistical model, its specification can be based exclusively on the conditional distribution D(y,/X,; ,) The same applies to the specification of the dynamic linear regression model which can be based exclusively on D(y,/Zp_,, X,3 W,) In such a case, however, certain restrictions need to be imposed on the parameters of the statistical generating mechanism (GM):

(see Chapter 8) It is important to note that in the case where {Z„,£€ T} is

assumed to be both stationary and asymptotically independent the above restriction on the roots of the polynomial in (17) is satisfied automatically For notational convenience let us rewrite the statistical GM (16) in the more concise form

where

ÿ* =(.#¿, „ o Ba}: m(k + 1) x T1,

XPS Vpn ae Ven X6 XS ccvXc-m) mHẾK + Ð) x T For the sample period t=m+1, , 7, (18) can be written in the following matrix form:

23.1 Specification $31 where y: (T—m) x 1, X*:(T—m) x m(k +1) Note that x, is k x | because it includes the constant as well but x,_;,i=1,2, ,m,are(k — 1) x 1 vectors; this convention is adopted to simplify the notation Looking at (18) and (19) the discerning reader will have noticed a purposeful attempt to use notation which relates the dynamic linear regression model to the linear and stochastic linear regression models Indeed, the statistical GM in (18) and (19) is a hybrid of the statistical GM’s of these models The part

3721 #/y,—¡ 1s directly related to the stochastic linear regression model in view of the conditioning on the o-field o(YP_,) and the rest of the systematic component being a direct extension of that of the linear regression model This relationship will prove very important in the statistical analysis of the parameters of the dynamic linear regression model discussed in what follows

In direct analogy to the linear and stochastic linear regression models we need to assume that X* as defined above is of full rank, ie rank(X*)= m(k + 1) for all the observable values of ¥?_;=(\ms Vmate +: +> Vr-a): The probability model underlying (16) comes in the form of the product of the sequentially conditional normal distribution D(y,/Z?_,, X35 Wy), t>m For the sample period t=m+1, , T the distribution of the sample is

can be ignored: a strategy adopted in Section 23.2 for expositional purposes The interested reader should consult Priestley (1981) for a readable discussion of how the initial conditions can be treated Further discussion of these conditions is given in Section 23.3 below

The dynamic linear regression model — specification

@®) The statistical GM

v= Box, + > XV, Š X,.;+u, t>m (23.22)

¿=1

see Appendix 22.1 for the form of the mapping Ø* =H(y,)

X, is strongly exogenous with respect to 6*

The parameters #=(a,,%>, , %,,)' Satisfy the restriction that all the roots of the polynomial

m-1

("- » ae i=1 are less than one in absolute value

rank(X*)=m(k + 1) for all observable values of Y?_,,, T>m(k+t 1)

The probability model

=< D(y,/Z)_ 1, X,5 0*) = Ú/%-¡: Xu 09) som) gạt B n) x5 (¥,-P*’X*)?>,

OF eRMETD x astm

(23.24)

(i) D(y,/Z?_,,X,; 6*) is normal;

(ii) E(y,/o(¥0_ ,),X2=x°)= p*’X* — linear in X>;

(ili) Var(y,/o(Y?_,), X°=x°)=o2 — homoskedastic (free of

Xử);

0* is time invariant

The sampling model

Y=(nst1> Vmt2: «++: JrÝ 1S a Stationary, asymptotically

independent sample sequentially drawn from D(y,/Z°_,,X,;0*), t=m+1,m+4+2, , T, respectively

Note that the above specification is based on D(4,/Z°_,, t-—1° X,; 0%), t=m+ 1,

, T, directly and not on D(Z,/Z?_,: w) This is the reason why we need

assumption [4] in order to ensure the asymptotic independence of

u 1 ca Sy), b> mh

23.2 Estimation 533 23.2 Estimation

In view of the probability and sampling model assumptions [6] to [8] the likelihood function is defined by

where i= y—X*f* The estimators B* and @ are said to be approximate

maximum likelihood estimators (MLE’s) of B* and a@, respectively, because the initial conditions have been ignored The formulae for these estimators bring out the similarity between the dynamic, linear and stochastic linear regression models Moreover, the similarity does not end with the formulae Given that the statistical GM for the dynamic linear regression model can

be viewed as a hybrid of the other two models we can deduce that in direct analogy to the stochastic linear regression model the finite sample

distributions of B* and 4; are likely to be largely intractable One important

difference between the dynamic and stochastic linear regression models, however, is that in the former case, although the orthogonality between yp, and u, (4, L u,} holds in the decomposition

One important implication of this is that B* is a biased estimator of B*, ice

E(B* — p*)= E[((X*"X*)'!X*'u] 40 (23.32) This, however, is only a problem fora small T because asymptotically 6* = (B*, G3) enjoys certain attractive properties under fairly general conditions, including asymptotic unbiasedness

a MLE of 6* can be deduced Let us consider the argument in more detail

If we define the information set A _, =(o(¥?_,), XP =x°)) we can deduce

that the non-systematic component as defined above takes the form

and the sequence {u,, #%, t>m} represents a martingale difference, ie Eu,/2, _¡)=0, t>m (see Section 8.4) Using the limit theorems related to martingales (see Section 9.3) we can then proceed to derive the asymptotic

23.2 Estimation 535 that

xứ as

Hence,

The convergence in (37) stems from the fact that (X#u, Z2; delines a

martingale difference given that

E(X*u,/F_,)=X*E(u,/F_,)=0, i=1,2, ,m(k+ 1) (23.38)

This suggests that the main assumption underlying the strong consistency

of B* is the order of magnitude of E(X*’X*) and the non-singularity of E(X*’X*/T) for T > m(k + 1) Given that X¥ =(),— 1, Vyas- +> Ve-ms Xto Xp 5X;—m) We can ensure that E(X*’X*) satisfies these conditions if the cross- products involved satisfy the restrictions As far as the cross-products which involve the y,_;8 are concerned the restrictions are ensured by assumption [4] on the roots of (A"— 5% ,'a,1"~'}=0 For the x,s we need to assume that:

(i) xi] <C, i= 1, 2, ,k, te T, C being a constant;

(ii) lim;.,„[1/(T—+)]}/=á2xx;.,=Q, exists for r> 1 and in particular

Q, is also non-singular

These assumptions ensure that E(X*’X*)=O(T),i.e.G;=O(1) and G, +G where G is non-singular These in turn imply that I,(0*)=O(7T) and 1,,(6*) = OW) which ensures that not only (37) holds but

The (weak) consistency of @* as an estimator of 6*, ie

Example

The tests for independence applied to the money equation in Chapter 22 showed that the assumption is invalid Moreover, the erratic behaviour of the recursive estimators and the rejection of the linearity and homoskedasticity assumptions in Chapter 21 confirmed the invalidity of the conditioning on the current observed values of X, only In such a case the natural way to proceed is to respecify the appropriate statistical model for the modelling of the money equation so as to take into consideration the

time dependence in the sample

In view of the discussion of the assumption of stationarity as well as economic theoretical reasons the dependent variable chosen for the postulated statistical GM is m* =In (M,/P,) (see Fig 19.2) The value of the maximum lag postulated is m=4, mainly because previous studies

demonstrated the optimum lag for ‘memory’ restriction adequate to

characterise similar economic time series (see Hendry (1980)) The

postulated statistical GM is of the form

me = Bot » am + » (Bi iV, it BaiPr—it Bait, -i)

i=0 i=1

+O Q1 6209, +630 31 + Ue (23.44) where Q,,, = 1, 2, 3, are three dummy variables which refer to important monetary policy changes which led to short-run ‘unusual’ changes:

(O,,-197 liv) The introduction of competition and credit control.

23.2 Estimation 537 Table 23.1 Estimated statistical GM

(Q5,-1975it) The suspension of the corset and the Bank of England asked

the banks to channel the new lending away from personal loans

(Q3,-1982i) The introduction of MI us a monetary target

The estimatc<! coefficients for the period 1964; 1982ir are shown in Table 23.1 Estirnation of (44) with Am# as the dependent variable changed only

the goodiess-of-fit measure as given by R? and R* to R?=0.796 and R*=

0.711 The change measures the loss of goodness of fit due to the presence of

a trend (compare Fig 19.2 with 21.2} The parameters 0=(fo ¡; a; Bs)

%¡+¡.Í=0, 1,2, 3.4, c¡, ca cy đa) in terms of which the statistical GM is defined are the stutistical and not the (economic) theoretical parameters of interest In order to be able to determine the latter (using specification testing), we need to ensure first that the estimated statistical GM is well defined That is that the assumptions [1] -{8] underlying the statistical model are indeed valid Testing for these assumptions is the task of misspecification testing in the context of the dynamic linear regression model considered in the next section

Looking at the time graph of the actual (y,) and fitted (1) (see Fig 23.1) values of the dependent variable we can see that 3, ‘tracks’ \, quite closely for the estimation period This is partly confirmed by the value of the variance of the regression which is nearly one-tenth of that of the money equation estimated in Chapter 19; see Fig 23.2 showing the two sets of residuals This reduction is even more impressive given that it has been achieved using the same sample information set In terms of the R* we can see that this also confirms the improvement in the goodness of fit being

more than double the original one with m as the dependent variable and y,,

p, and i, only the regressors (see Chapter 19)

Before we proceed with the misspecification testing of the above estimated statistical GM it is important to comment on the presence of the

23.3 Misspecification testing 539 three dummy variables Their presence brings out the importance of the

sample period economic ‘history’ background in econometric modelling

Unless the modeller is aware of this background the modelling can very easily go astray In the above case, leaving the three dummies out, the normality assumption will certainly be affected It is also important to remember that such dummy variables are only employed when the

modeller believes that certain events had only a temporary effect on the

relationship without any lasting changes in the relationship In the case of longer-term effects we need to model them, not just pick them up using dummies Moreover, the number of such dummies should be restricted to

be relatively small A liberal use of dummy variables can certainly achieve wonders in terms of goodness of fit but very little else Indeed, a dummy variable for each observation which yields a perfect fit but no ‘explanation’ of any kind In a certain sense the coefficients of dummy variables represent a measure of our ‘ignorance’

23.3 Misspecification testing

As argued above, specification testing is based on the assumption of correct specification, that is, the assumptions underlying the statistical model in question are valid This is because departures from these assumptions can invalidate the testing procedures For this reason it is important to test for the validity of these assumptions before we can proceed to determine an empirical econometric model on the sound basis of a well-defined estimated statistical GM

(1) Assumption underlying the statistical GM

Assumption [1] refers to the definition of the systematic and non-systematic components of the statistical GM The most important restriction in defining the systematic component as

(i) m chosen ‘too large

If m is chosen larger than its optimum but unknown value m* then ‘near

collinearity’ (insufficient data information) or even exact collinearity will

‘creep’ into the statistical GM (see Section 20.6) This is because as m is increased the same observed data are ‘asked’ to provide further and further information about an increasing number of unknown parameters The implications of insufficient data information discussed in the context of the linear regression problem can be applied to the DLR model with some reinterpretation due to the presence of lagged y,s m X*

{it} m chosen ‘too small

If mis chosen ‘too small’ then the omitted lagged Z,s will form part of the unmodelled part of y, and the error term

ux = Vt ~ Box, ~— x (4y; —ỉ + Bx, -;) (23.46)

i=1

is no longer non-systematic relative to the information set

That is, {u*,.F,,t>m} will no longer be a martingale difference, having very

serious consequences for the properties of 6* discussed in Section 23.2 In particular the consistency and asymptotic normality of @* are no longer valid, as can be verified using the results of Section 22.1; see also Section 23.2 above Because of these implications it is important to be able to test for m<m*, Given that the ‘true’ statistical GM is given by

This implies that m<m* can be tested using the null hypothesis

Hgẹ:zt—=f and p*=0 against H,:a%40 or Ø*z0,

where

Min = (Sm eases Ln)’ Bà =(Êm : v - - ‹ Ủm*)-

The obvious test for such a hypothesis will be analogous to the F-type test for independence suggested in Chapter 22 Alternatively, we could use an asymptotically equivalent test based on TR? from the auxiliary regression

23.3 Misspecification testing 541 where * refers to the residuals from the estimation of

(see Chapter 22 for more details)

The F-type test for the money equation estimated in Section 23.3 with m* =6 yielded

with certain modifications as indirect tests of m<m* As far as the

asymptotic Lagrange multiplier (LM) tests for AR(p) or MA(p), where p> 1, based on the auxiliary regressions are concerned, can be applied in the present context without any modifications The reason is that the

presence of the lagged y,s in the systematic component of the statistical GM

(51) makes no difference asymptotically On the other hand, the Durbin— Watson (DW) test is not applicable in the present context because the test depends crucially on the non-stochastic nature of the matrix X Durbin (1970) proposed a test for AR(1) (u* = pu*_, + u,) in the context of the DLR

model based on the so-called h-test statistic defined by

dependency (AR(1) or MA(1)) takes the form

Asin the case of the linear regression model the above Durbin’s h-test and the LM({J) (I> 1) test can be viewed as tests of significance in the context of the auxiliary regression

t

i=1

The obvious way to test

Ho: py =P2.='°° =p, =0, H,:p,#0, for any i=1,2, ,1

(23.59)

is to use the F-test approximation which includes the degrees of freedom correction term instead of its asymptotic chi-square form The autocorrelation error tests will be particularly useful in cases where the F- test based on (48) cannot be applied because the degrees of freedom are at a premium

In the case of the estimated money equation in Table 23.1 the above test statistics for «=0.05 yielded:

As we can see from the above results in all cases the null hypothesis 1s

accepted confirming (53) above

23.3 Misspecification testing 343 The above tests can be viewed as indirect ways to test the assumption postulating the adequacy of the maximum lag m The question which naturally arises is whether m can be determined directly by the data In the statistical time-series literature this question has been considered extensively and various formal procedures have been suggested such as Akaike’s AIC and BIC or Parzen’s CAT criteria (see Priestley (1981) for a readable summary of these procedures) In econometric practice, however,

it might be preferable to postulate m on a priori grounds and then use the above indirect tests for its adequacy

Assumption [2] specifies the statistical parameters of interest as being 0* = (1, -s Oms Bor Bis «+++ Bm» 0) These parameters provide us with an opportunity to consider two issues we only discussed in passing The first issue is related to the distinction made in Chapter 17 between the statistical and (economic) theoretical parameters of interest In the present context 6*

as defined above has very little, if any, economic interpretation Hence, 6* represents the statistical parameters of interest These parameters enable us

to specify a well-defined statistical model which can provide the basis of the

‘design’ for an empirical econometric model As argued in Chapter 17, the estimated statistical GM could be viewed as a sufficient statistic for the theoretical parameters of interest The statistical parameters of interest provide only a statistically ‘adequate’ (sufficient) parametrisation with the theoretical parameters of interest being defined as functions of the former This is because a theoretical parameter is well defined (statistically) only when it is directly related to a well-defined statistical parameter The determination of the theoretical parameters of interest will be considered in Section 23.4 on specification testing The second related issue is concerned with the presence of ‘near’ collinearity In Section 20.6 it was argued that

‘near’ collinearity is defined relative to a given parametrisation and information set In the present context it is likely that ‘near’ collinearity (or insufficient data information) might be a problem relative to the parametrisation based on @* The problem, however, can be easily overcome

in determining the theoretical parameters of interest so as to ‘design’ a parsimonious as well as ‘robust’ theoretical parametrisation Both issues will be considered in Section 23.6 below in relation to the statistical GM estimated in Section 23.2

Assumption [3] postulates the strong exogeneity of X, with respect to the parameters 6* fort=m-+1 T As far as the weak exogeneity component

of this assumption is concerned it will be treated as a non-testable presupposition as in the context of the linear regression model (see Section 20.3) The Granger non-causality component, however, is testable in the context of the general autoregressive representation

where g(t)=c, A, $ C245 + °°: +644, (see Dhrymes (1978)) The component

23.3 Misspecification testing 545 g(t), called the complementary function, is the solution of the homogeneous difference equation o(L)y,=O and c=(c, C3, ., C,) are constants determined by the initial conditions y,, ¥3, Vy, Via

Vp Hey tent +e,

Va HCqAy HegdAg tc $e yA,

Vm =a eg ag te teem!

In order to ensure the asymptotic independence (stationarity) of

‘y,,t€ 1} we need this component to decay to zero as f > x in order for {y,,t€ 1} to ‘forget’ the initial conditions (see Priestley (198 1)) For this to

be the case (4, 43,.-., 4), which are the roots of the polynomial

a A) = (A — a amt — +++ — a) =0 (23.70)

should satisfy the restrictions

Note that the roots of z(L) are (1/2;),?= Í,2, m

When the restrictions hold

t7rx

As argued in Section 23.1 in the case where {Z,,t¢ 1} is assumed to be a

stationary, asymptotically independent process at the outset, the time

invariance of the statistical parameters of interest and the restrictions

|2|<1.¡=1.2 m are automatically satisfied

In order to get some idea as to what happens in the case where the restrictions |/,|<1.i=1.2 m, are not satisfied let us consider the simplest case where m= 1 and 4, =1 (x, =1), Le

t

Ve ¥p-4 +, +u,= » (w,+u,), (23.73)

s=1

These suggest that both the mean and covariance of { y,,t€ T} increase with

tand thus as tf > « they become unbounded Thus {y,, te T}, as generated

by (73), is not only non-stationary (with its mean and covariance varying

with t) but also its ‘memory’ remains constant.

In the case where |x,|> 1 again {y,,t¢€T} is non-stationary since

and thus E(y,) +o and Cov(y,y,,,) 70 as to Moreover, the

‘memory’ of the process increases as the gap t> w!

In the simplest case where m=1 when the restriction |/,|<1 is not

satisfied we run into problems which invalidate some of the results of Section 23.2 In particular the asymptotic properties of the approximate MLE’s of 6* need to be modified (see Fuller (1976) for a more detailed discussion) For the general case where m> 1 the situation is even more complicated and most of the questions related to the asymptotic properties

of 6* are as yet unresolved (see Rao (1984)

Assumption [5], relating to the rank of X* =(),~4, + Wy—m Xe X£T 1a Xi), has already been discussed in relation to assumption [1] In the case where for the observed sample y the rank condition falls, i.e

a unique 6* does not exist If this is due to the fact that the postulated m is

much larger than the optimum maximum lag m* then the way to proceed is

to reduce m If, however, the problem is due to the rank of the submatrix

X =(X,, X2, , X7) then we need to reparametrise (see Chapter 20) In either case the problem is relatively easy to detect What is more difficult to detect is ‘near’ collinearity which might be particularly relevant in the present context As argued above, however, the problem is relative to a given parametrisation and thus can be tackled alongside the

reparametrisation of (48) in our attempt to ‘design’ an empirical

econometric model based on the estimated form of (48) (see Section 20.6)

(2) Assumptions underlying the probability model

The assumptions underlying the probability model constitute a hybrid of those of the linear and stochastic linear regression models considered in

Chapters 19 and 20 respectively The only new feature in the present context

is the presence of the initial conditions coming in the form of the following

distribution:

D(Z¡: $) [[ D(y//22—¡.X„ 9) (23.77)

t=2

where @ = H() For expositional purposes we chose to ignore these initial

conditions in Sections 23.1 and 23.2 This enabled us to ignore the problem

for the period t=1, 2, , m The easiest way we can take the initial

conditions into consideration is to assume that @ coincides with @* If this is

adopted the approximate MLE’s &* will be modified in so far as the various summations involved will no longer be of the form }'/_,, ,, uniformly but

a1 aX*X*; and V7, ,X* ,y,, i=1, 2, , m That is, start summing

from the point where observations become available for each individual

component

As far as the assumptions of normality, linearity and homoskedasticity

are concerned the same comments made in Chapter 21 in the context of the

linear regression model apply here with minor modifications In particular the results based on asymptotic theory arguments carry over to the present

case The implications of non-normality, as defined in the context of the

linear regression model (see Chapter 21), can be extended directly to the

DLR model unchanged The OLS estimators of B* and a4 have more or less

the same asymptotic properties and any testing based on their asymptotic distributions remains valid In relation to misspecification testing for departures from normality the asymptotic test based on the skewness and kurtosis coefficients remains valid without any changes Let us apply this

test to the money equation estimated in Section 23.3 The skewness—

kurtosis test statistic is t(y) = 2.347 which implies that for « =0.05 the null hypothesis of normality is not rejected since c, = 5.99

Testing linearity in the present context presents us with additional problems in so far as the test based on the Kolmogorov—Gabor polynomial (see (21.11) and (21.77)) will not be operational The RESET type test, however, based on (21.10) and (21.83) is likely to be operational in the

present context Applying this test with }#* ($? and $3 were excluded because

of collinearity) in the auxiliary regression

hand, it might be interesting to use only the cross-products of the x,,s only

as in the linear regression case Such a test can be used to refute the

conjecture made in Chapter 21 about the likely source of the detected heteroskedasticity It was conjectured that invalid conditioning (ignoring