Real Estate Modelling and Forecasting Hardcover_14 pptx

In both equations, the error correction term takes anegative sign, indicating the presence of forces to move the relationshipback to equilibrium, and it is significant at the 1 per cent l

There are similarities but also differences between the two error tion equations above In both equations, the error correction term takes anegative sign, indicating the presence of forces to move the relationshipback to equilibrium, and it is significant at the 1 per cent level For the rent-GDP equation (12.56), the adjustment to equilibrium is 6.5 per cent everyquarter – a moderate adjustment speed This is seen in figure 12.8, wheredisequilibrium situations persist for long periods For the rent–employmenterror correction equation (12.57), the adjustment is higher at 11.8 per centevery quarter – a rather speedy adjustment (nearly 50 per cent every year).

correc-An interesting finding is that GDP is highly significant in equation (12.56), whereas EMP in equation (12.57) is significant only at the 10 per

cent level Equation (12.56) has a notably higher explanatory power with

an adjusted R2of 0.68, compared with 0.30 for equation (12.57) The results

of the diagnostic checks are broadly similar Both equations have residualsthat are normally distributed, but they fail the serial correlation tests badly.Serial correlation seems to be a problem, as the tests show the presence ofserial correlation for orders 1, 2, 3 and 4 (results for orders 1 and 4 only arereported here) Both equations fail the heteroscedasticity and RESET tests

An option available to the analyst is to augment the error correction tions and attempt to rectify the misspecification in the equations (12.56)and (12.57) in this way We do so by specifying general models containing

equa-four lags of GDP in equation (12.56) and equa-four lags of EMP in equation

(12.57) We expect this number of lags to be sufficient to identify the impact

of past GDP or employment changes on rental growth We subsequently

remove regressors using as the criterion the minimisation of AIC The GDP and EMP terms in the final model should also take the expected positive

signs For brevity, we now focus on the GDP equation

RENT t = −3.437 − 0.089RESGDPt−1+ 1.642GDPt−1+ 2.466GDPt−4

(−10.07) (−4.48) (2.23) (3.32) (12.58)

Adj R2= 0.69; DW = 0.43; number of observations = 66 (3Q1991–4Q2007) Diagnostics: normality BJ test value: 2.81 (p = 0.25); LM test for serial correlation (first order): 41.18 (p = 0.00); LM test for serial correlation (fourth order): 45.57 (p = 0.00);

heteroscedasticity with cross-terms: 23.43 (p = 0.01); RESET: 1.65 (p = 0.20).

Equation (12.58) is the new rent-GDP error correction equation The GDP

term has lost some of its significance compared with the original equation,and the influence of changes in GDP on changes in real rents in the presence

of the error correction term is best represented by the first and fourth lags

of GDP The error correction term retains its significance and now points

to a 9 per cent quarterly adjustment to equilibrium In terms of diagnostics,the only improvement made is that the model now passes the RESET test.

We use the above specification to forecast real rents in Sydney We carry

out two forecasting exercises – ex post and ex ante – based on our own assumptions for GDP growth For the ex post (out-of-sample) forecasts, we

estimate the models up to 4Q2005 and forecast the remaining eight quarters

of the sample Therefore the forecasts for 1Q2006 to 4Q2007 are produced

by the coefficients estimated using the shorter sample period (ending in4Q2005) This error correction model is

 RENTˆ t = −3.892 − 0.097RESGDPt−1+ 1.295GDPt−1

(−11.40) (−5.10) (1.87)

(4.31) Adj R2= 0.76; DW = 0.50; number of observations = 58 (3Q1991–4Q2005).

We can highlight the fact that all the variables are statistically significant,

with GDPt−1at the 10 per cent level and not at the 5 per cent level, whichwas the case in (12.58) The explanatory power is higher over this sampleperiod, which is not surprising given the fact that the full-sample modeldid not replicate the changes in rents satisfactorily towards the end of thesample Table 12.4 contains the forecasts from the error correction model.The forecast for 1Q2006 using equation (12.59) is given by

 RENTˆ 1Q2006 = −3.892 − 0.097 × (−7.06) + 1.295 × 0.5 + 3.043 × 0.2

This is the predicted change in real rent between 4Q2005 and 1Q2006, fromwhich we get the forecast for real rent for 1Q2006 of 82.0 (column (ii)) and thegrowth rate of−2.32 per cent (quarter-on-quarter [qoq] percentage change),shown in column (vii) The value of the error correction term in 4Q2005 isproduced by the long-run equation estimated for the shorter sample period(2Q1990 to 4Q2005):

ˆ

(−0.65) (7.42) Adj R2= 0.47; DW = 0.04; number of observations = 63 (2Q1990–4Q2005).

Again, we perform unit root tests on the residuals of the above equation.The findings reject the presence of a unit root, and we therefore proceed

to estimate the error correction term for 4Q2005 In equation (12.61), thefitted values are given by the expression (−7.167 + 0.642 × GDPt) The error

Table 12.4 Ex post forecasts from error correction model

Notes: Bold numbers indicate model-based forecasts ECT is the value of the error

correction term (the residual).

correction term is

ECT t = actual rent – fitted rent = RENTt − (−7.167 + 0.642GDPt)

= RENT t + 7.167 − 0.642GDPt Hence the value of ECT 4Q2005, which is required for the forecast of changes

Table 12.5 Forecast evaluation

Mean error 1.18Absolute error 1.37

Theil’s U1 statistic 0.61

Table 12.6 Ex ante forecasts from the error correction model

Notes: Bold numbers indicate forecasts The forecast assumption is that GDP grows at

0.5 per cent per quarter.

In 2007 the forecasts improved significantly in terms of average error.The ECM predicts average growth of 0.60, which is quite short of the actualfigure of 1.4 per cent per quarter We now use the model to forecast out eightquarters from the original sample period We need exogenous forecasts forGDP, and we therefore assume quarterly GDP growth of 0.5 per cent for theperiod 1Q2008 to 4Q2009 Table 12.6 presents these forecasts

For the ECM forecasts given in table 12.6, the coefficients obtained fromthe error correction term represented by equation (12.61) and the short-runequation (12.59) are used The ECM predicts a modest acceleration in realrents in 2008 followed by a slowdown in 2009 These forecasts are, of course,based on our own somewhat arbitrary assumptions for GDP growth

12.7 The Engle and Yoo three-step method

The Engle and Yoo (1987) three-step procedure takes its first two steps fromEngle–Granger (EG) Engle and Yoo then add a third step, giving updatedestimates of the cointegrating vector and its standard errors The Engle andYoo (EY) third step is algebraically technical and, additionally, EY suffersfrom all the remaining problems of the EG approach There is, arguably, a farsuperior procedure available to remedy the lack of testability of hypothesesconcerning the cointegrating relationship: the Johansen (1988) procedure.For these reasons, the Engle–Yoo procedure is rarely employed in empiricalapplications and is not considered further here

12.8 Testing for and estimating cointegrating systems using

the Johansen technique

The Johansen approach is based on the specification of a VAR model Suppose

that a set of g variables (g≥ 2) are under consideration that are I(1) and

that it is thought may be cointegrated A VAR with k lags containing these

variables can be set up:

of the matrix can be interpreted as a long-run coefficient matrix, since,

in equilibrium, all the yt − i will be zero, and setting the error terms, ut, to

their expected value of zero will leave yt − k= 0 Notice the comparabilitybetween this set of equations and the testing equation for an ADF test,which has a first-differenced term as the dependent variable, together with

a lagged levels term and lagged differences on the RHS

The test for cointegration between the ys is calculated by looking at the rank of the matrix via its eigenvalues.3The rank of a matrix is equal to thenumber of its characteristic roots (eigenvalues) that are different from zero

(see section 2.7) The eigenvalues, denoted λi, are put in ascending order:

λ1≥ λ2 ≥ · · · ≥ λg If the λs are roots, in this context they must be less than one in absolute value and positive, and λ1will be the largest (i.e the closest

to one), while λgwill be the smallest (i.e the closest to zero) If the variables

are not cointegrated, the rank of will not be significantly different from zero, so λi ≈ 0 ∀ i The test statistics actually incorporate ln(1 − λi), rather

than the λi themselves, but, all the same, when λi = 0, ln(1 − λi)= 0

Suppose now that rank ( ) = 1, then ln(1 − λ1)will be negative and ln(1−

λ i)= 0 ∀ i > 1 If the eigenvalue i is non-zero, then ln(1 − λi ) < 0 ∀ i > 1 That

is, for to have a rank of one, the largest eigenvalue must be significantly

non-zero, while others will not be significantly different from zero

There are two test statistics for cointegration under the Johansenapproach, which are formulated as

be eigenvectors A significantly non-zero eigenvalue indictates a significantcointegrating vector

λ trace is a joint test in which the null is that the number of cointegrating

vectors is smaller than or equal to r against an unspecified or general alternative that there are more than r It starts with p eigenvalues, and then, successively, the largest is removed λ trace = 0 when all the λi = 0, for

i = 1, , g.

λ maxconducts separate tests on each eigenvalue, and has as its null

hypoth-esis that the number of cointegrating vectors is r against an alternative of

r+ 1

moment matrices and not from itself.

Johansen and Juselius (1990) provide critical values for the two statistics.The distribution of the test statistics is non-standard, and the critical values

depend on the value of g − r, the number of non-stationary components

and whether constants are included in each of the equations Intercepts can

be included either in the cointegrating vectors themselves or as additionalterms in the VAR The latter is equivalent to including a trend in the data-generating processes for the levels of the series Osterwald-Lenum (1992)and, more recently, MacKinnon, Haug and Michelis (1999) provide a morecomplete set of critical values for the Johansen test

If the test statistic is greater than the critical value from Johansen’s tables,

reject the null hypothesis that there are r cointegrating vectors in favour of the alternative, that there are r + 1 (for λtrace ) or more than r (for λ max) The

testing is conducted in a sequence and, under the null, r = 0, 1, , g − 1,

so that the hypotheses for λmaxare

The first test involves a null hypothesis of no cointegrating vectors

(corre-sponding to having zero rank) If this null is not rejected, it would be

concluded that there are no cointegrating vectors and the testing would

be completed If H0: r = 0 is rejected, however, the null that there is onecointegrating vector (i.e H0: r = 1) would be tested, and so on Thus the

value of r is continually increased until the null is no longer rejected How does this correspond to a test of the rank of the matrix, though?

r is the rank of cannot be of full rank (g) since this would correspond

to the original yt being stationary If has zero rank then, by analogy to the univariate case, yt depends only on yt − j and not on yt−1, so that there

is no long-run relationship between the elements of yt−1 Hence there is nocointegration

For 1 < rank( ) < g, there are r cointegrating vectors is then defined

as the product of two matrices, α and β, of dimension (g × r) and (r × g),

respectively – i.e

The matrix β gives the cointegrating vectors, while α gives the amount of

each cointegrating vector entering each equation of the VECM, also known

as the ‘adjustment parameters’

For example, suppose that g= 4, so that the system contains four

vari-ables The elements of the matrix would be written

Given (12.73), it is possible to write out the separate equations for each

variable yt It is also common to ‘normalise’ on a particular variable, sothat the coefficient on that variable in the cointegrating vector is one For

example, normalising on y would make the cointegrating term in the

12.8.1 Hypothesis testing using Johansen

The Engle–Granger approach does not permit the testing of hypotheses

on the cointegrating relationships themselves, but the Johansen set-updoes permit the testing of hypotheses about the equilibrium relationshipsbetween the variables Johansen allows a researcher to test a hypothesisabout one or more coefficients in the cointegrating relationship by viewing

the hypothesis as a restriction on the matrix If there exist r cointegrating

vectors, only those linear combinations or linear transformations of them,

or combinations of the cointegrating vectors, will be stationary In fact, the

matrix of cointegrating vectors β can be multiplied by any non-singular

conformable matrix to obtain a new set of cointegrating vectors

A set of required long-run coefficient values or relationships betweenthe coefficients does not necessarily imply that the cointegrating vectorshave to be restricted This is because any combination of cointegratingvectors is also a cointegrating vector It may therefore be possible to combinethe cointegrating vectors thus far obtained to provide a new one, or, ingeneral, a new set, having the required properties The simpler and fewerthe required properties are, the more likely it is that this recombination

process (called renormalisation) will automatically yield cointegrating vectors

with the required properties As the restrictions become more numerous orinvolve more of the coefficients of the vectors, however, it will eventuallybecome impossible to satisfy all of them by renormalisation After this point,all other linear combinations of the variables will be non-stationary If therestriction does not affect the model much – i.e if the restriction is notbinding – then the eigenvectors should not change much following theimposition of the restriction A statistic to test this hypothesis is given by

where λ∗i are the characteristic roots of the restricted model, λi are the

characteristic roots of the unrestricted model, r is the number of zero characteristic roots in the unrestricted model and m is the number of

non-restrictions

Restrictions are actually imposed by substituting them into the relevant

α or β matrices as appropriate, so that tests can be conducted on either the

cointegrating vectors or their loadings in each equation in the system (orboth) For example, considering (12.69) to (12.71) above, it may be that theorysuggests that the coefficients on the loadings of the cointegrating vector(s)

in each equation should take on certain values, in which case it would be

relevant to test restrictions on the elements of α – e.g α11= 1, α23= −1,etc Equally, it may be of interest to examine whether only a subset of the

variables in ytis actually required to obtain a stationary linear combination

In that case, it would be appropriate to test restrictions of elements of β For example, to test the hypothesis that y4is not necessary to form a long-run

relationship, set β14= 0, β24= 0, etc For an excellent detailed treatment ofcointegration in the context of both single-equation and multiple-equationmodels, see Harris (1995)

12.9 An application of the Johansen technique to securitised

real estate

Real estate analysts expect that greater economic and financial market ages between regions will be reflected in closer relationships between mar-kets The increasing global movements of capital targeting real estate fur-ther emphasise the connections among real estate markets Investors, intheir search for better returns away from home and for greater diversifica-tion, have sought opportunities in international markets, particularly in themore transparent markets (see Bardhan and Kroll, 2007, for an account ofthe globalisation of the US real estate industry) The question is, of course,whether the stronger economic and financial market dependencies andglobal capital flows result in greater integration between real estate mar-kets and, therefore, stronger long-run relationships

link-We apply the Johansen technique to test for cointegration between threecontinental securitised real estate price indices for the United States, Asiaand Europe For the global investor, these indices could represent oppor-tunities for investment and diversification They give exposure to differentregional economic environments and property market fundamentals (forexample, trends in the underlying occupier markets) Given that these arepublicly traded indices, investors can enter and exit rapidly, so it is a liq-uid market This market may therefore present arbitrage opportunities toinvestors who can trade them as expectations change Figure 12.9 plots thethree indices

130 Jan 90=100 Asia Europe United States

120

110 100

90 80

Jan 90 Jan 91 Jan 92 Jan 93 Jan 94 Jan 95 Jan 96 Jan 97 Jan 98 Jan 99 Jan 00 Jan 01 Jan 02 Jan 03 Jan 04 Jan 05 Jan 06 Jan 07

in the other regions whereas the fall of the Asian index in 1998 and 1999reflected the regional turbulence (the currency crisis) Figure 12.10 plots thereturns series

online databases.

It is clear from plotting the series in levels and in first differences that theywill have a unit root in levels but in first differences they look stationary.This is confirmed by the results of the ADF tests we present in table 12.7.The ADF tests are run with a maximum of six lags and AIC is used to selectthe optimal number of lags in the regressions In levels, all three serieshave unit roots In first differences, the hypothesis of a unit root is stronglyrejected in all forms of the ADF regressions.

The Johansen technique we employ to study whether the three securitisedprice indices are cointegrated implies that all series are treated as endoge-nous Table 12.8 reports the results of the Johansen tests The empiricalanalysis requires the specification of the lag length in the Johansen VAR

We use AIC for the VAR system to select the optimum number of lags Wespecified a maximum of six lags and AIC (value= 6.09) selected two lags in

the VAR

Both the λmax and the λtracestatistics give evidence, at the 10 per cent and

5 per cent levels, respectively, of one cointegrating equation The maximum

eigenvalue λmax and λtrace statistics reject the null hypothesis of no

cointe-gration (r = 0), since the statistic values are higher than the critical values

at these levels of significance, in favour of one cointegrating vector (r = 1).These tests do not reject the null hypothesis of a cointegrating vector (teststatistic values lower than critical values) On the basis of these results, it isconcluded that the European, Asian and US indices exhibit one equilibriumrelationship and that they therefore move in proportion in the long run.The cointegrating combination is given by

87.76 + ASIA − 4.63US + 3.34EU

and a plot of the deviation from equilibrium is presented in figure 12.11

We observe long periods when the error correction term remains wellabove or well below the zero line (which is taken as the equilibrium path),although quicker adjustments are also seen on three occasions This errorcorrection term is not statistically significant in all short-term equations,

however, as the VECM in table 12.9 shows (with t-ratios in parentheses).

For both Europe and Asia, the coefficient on the error correction term isnegative, whereas, in the US equation, it is positive When the three seriesare not in equilibrium, the Asian and European indices adjust in a similardirection and the US index in the opposite direction The only significanterror correction term in the VECM is that in the European equation, how-ever Hence the deviations from the equilibrium path that these indicesform are more relevant in determining the short-term adjustments of theEuropean prices than in the other markets The coefficients on the errorcorrection term point to very slow adjustments of less than 1 per cent each

Table 12.7 Unit root tests for securitised real estate price indices

Unit roots in price index levels (AS, EU, US) Unit roots in first differences (AS, EU, US)

Table 12.8 Johansen tests for cointegration between Asia, the United States and

Europe

Null Alt statistic (p-value) Null Alt statistic (p-value)

Notes: Lags = 2 r is the number of cointegrating vectors The critical values are taken

from MacKinnon, Haug and Michelis (1999).

70 60 50 40 30 20 10 0

−10

−20

−30 Jan 90 Jan 91 Jan 92 Jan 93 Jan 94 Jan 95 Jan 96 Jan 97 Jan 98 Jan 99 Jan 00 Jan 01 Jan 02 Jan 03 Jan 04 Jan 05 Jan 06 Jan 07

The explanatory power of the Asian equation is zero, with all variables

statistically insignificant The US equation has an adjusted R2 of 6 percent, with only the first lag of changes in the European index being sig-

nificant The European equation explains a little more (adjusted R2 = 15%)

with, again, EUt−1the only significant term It is worth remembering, ofcourse, that the monthly data in this example have noise and high volatil-ity, which can affect the significance of the variables and their explanatorypower

If we rearrange the fitted cointegrating vector to make Asia the subject

of the formula, we obtainASI Aˆ = −87.76 + 4.63US − 3.34EU If we again

define the residual or deviation from equilibrium as the difference between

Table 12.9 Dynamic model (VECM)

the actual and fitted values, then ˆu t = ASIAt− ˆASIA t Suppose that ˆu t >0, as

it was in the early 1990s (from January 1990 to the end of 1991), for example,then Asian real estate is overpriced relative to its equilibrium relationshipwith the United States and Europe To restore equilibrium, either the Asianindex will fall, or the US index will rise or the European index will fall

In such circumstances, the obvious trading strategy would be to buy USsecuritised real estate while short-selling that of Asia and Europe

The VECM of table 12.9 is now used to forecast Broadly, the steps aresimilar to those for the Engle–Granger technique and those for VAR fore-casting combined The out-of-sample forecasts for six months are given intable 12.10

The forecasts in table 12.10 are produced by using the three short-termequations to forecast a step ahead (August 2007), using the coefficientsfrom the whole-sample estimation, and subsequently to estimate the errorcorrection term for August 2007 The computations follow:

AS Aug−07= 0.0624 − 0.0016 × (−5.44) + 0.0832 × (−0.41) − 0.0442

× (−0.85) + 0.0576 × (−1.36) + 0.0910 × (−1.62) + 0.1267

× (−1.09) − 0.1138 × (−1.40) = −0.13

Table 12.10 VECM ex ante forecasts

Asia Europe United States AS EU US ECT

Apr 07 114.43 112.08 126.81May 07 115.05 112.06 126.78Jun 07 114.20 110.66 125.16 −0.85 −1.40 −1.62 −7.69Jul 07 113.79 109.57 123.80 −0.41 −1.09 −1.36 −5.44

forecasts in real time, the model would need to be run using data until January 2007 only.