Serial correlation and seasonality in the real estate market

Journal of Real Estate Finance and Economics, 12:139-162 (1996) �9 1996 Kluwer Academic Publishers

Serial Correlation and Seasonality in the Real Estate Market

CHIONG-LONG KUO Office of Government Services, Price Waterhouse LLP,, 1616 N. Fort Myer Dr., Arlington, VA 22209

Abstract

In this article, a two-step, two-sample method and a Bayesian method are proposed to estimate the serial correlation and the seasonality of the price behavior of the residential housing market. The Bayesian method is found to be superior to the alternative two-step methods. The empirical results based on the Bayesian approach support the rejection of the random-walk hypothesis in the real estate market. Seasonality is not significant; however, there is still a clear indication that the returns associated with seasonal dummies are strongest in the second quarter, with the first quarter following closely.

Key Words:

Due to transaction costs and taxes, the market for houses is often assumed to be less efficient than the markets for more liquid financial assets. Real estate prices may not follow a random-walk pattern. One way to test the random-walk hypothesis is to estimate the autocorrelation and to test whether the autocorrelation is significantly different from zero. Furthermore, an accurate measure of the serial correlation can be used to predict future price movement and would be useful for homeowners and real estate investors.

When asset prices are observed in each period, as in the stock market, the estimators for the return autocorrelation parameters obtained from running an OLS autoregressive regression are consistent, although not unbiased, In real estate markets, properties are traded infrequently, and an index or return series must be constructed to measure the overall price movement. The constructed return series is contaminated with errors, and the errors con- mined in the returns of different periods may be correlated. I f one simply treats the error- ridden return series as true returns and performs the autoregressive regression, the autocorrelation estimator would be inconsistent. Using repeat-sales data to test the efficiency of the residential housing markets in four cities, Case and Shiller (1989) observe this problem and propose a data-partition method to estimate the serial correlation. As discussed in the next section, their estimator is still not consistent and involves an arbitrary partition of the data set.

This article addresses the econometric problem associated with the estimation of serial correlation and seasonality, using repeat-sales data in the real estate market. Two methodologies, a two-step, two-sample method and a Bayesian method, are proposed to solve this problem. Unlike existing two-step methods, which use the indexes estimated in the first step to estimate serial correlation in the second step, the Bayesian method treats

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the underlying indexes as random variables. Moreover, the Bayesian method does not arbitrarily partition the data set. Therefore, the Bayesian method has advantages over the alternative two-step methods.

The remainder of the article is organized as follows. The next section discusses the existing two-step procedures and proposes a consistent two-step estimator for estimating the serial correlation of real estate market returns. Section 2 develops a Bayesian methodology to test and estimate serial correlation. Section 3 applies the two-step and Bayesian methodologies to repeat-sales data in Atlanta, Chicago, Dallas, and San Francisco from 1971 to 1986. In section 4, the Bayesian method is modified to take seasonality into account. Section 5 concludes the article.

1. Two-step methods for estimating serial correlation

Two-step methods first construct real estate price indexes, and then use the constructed index series to estimate serial correlation by running an autoregressive regression. Case and Shiller (1989) use repeat-sales data of single-family homes and employ the Weighted Repeated Sales (WRS) indexes from Case and ShiUer (1987) to test the random-walk hypothesis in the real estate market. They observe that the change in the estimated index cannot be regressed on the lagged change in the same index series for the purpose of testing the random-walk property, because the returns of different periods are contaminated with errors, which may be correlated. If this regression is performed, the dependent and independent variables will contain the same noise, and the estimator will be biased and inconsistent.

For example, let Yit denote the log price of house i at time t. If the data include only Yll, Y13, Y22, and Y23, then the estimated first-period return is (Y13 - Yll) - (Y23 - Y22), and the second-period return is (Y23 - Y22). Hence, the return in the first period is nega- tively correlated with the return in the second period. The correlation can be positive as well (see Case and Shiller, 1987, pp. 127-128, for further details). This problem occurs whether one uses the Baith-Muth-Nourse (BMN) index or the WRS index.

To cope with this problem, Case and Shiller randomly partition the data into two samples and construct one index series from each sample. Both series of indexes are still contaminated with noise, but the noise term in one series is not correlated with that in the other series. They regress the change of one index series on the lagged change of the other index series to estimate the serial correlation of index returns. Their results show that the annual serial correlations are significantly positive at the 5 % level for Chicago and San Francisco. Their effort to solve the inconsistency problem by splitting the data, however, is incomplete, and the serial correlation estimator is still inconsistent.

To see this, suppose that the change of the log index from time t - 1 to time t, Ot,

follows an AR(1) process,

0 t - - POt_ 1 + e t ( t = 2 . . . . . T + 1), (1)

where et is an i.i.d, noise term and is independent of 0s for all s and t. Then # = pl im(O~O/T) /p l im(O~OL/T) , where 0 = ( 0 2 . . . . . 0r+l)' and 0/; = (01 . . . . . 0r) '. Denote the

SERIAL CORRELATION AND SEASONALITY 141

return series (i.e., change in the index series) estimated from the full data set by O and its one-period lag by 0L, which can be written as 0 = 0 + 7/and 0L = ,0L + ~/L, respectively. The OLS estimator for the AR(1) regression coefficient, ~ = (OLO")/(O~OL), is inconsistent, since 7/ and 7/L are correlated. By partitioning the data into two samples, we can obtain two return series, 0a (lag-one-period return, denoted as 0~) and 0B (lag-one- period return, denoted as 0nL). We can write these two return series and their lags as 0a = 0 + ~/a, 0AL = 0L + ~ , 0S = 0 + ~B, and 0B/~ = 0L + ~sL, and assume that all the four error terms are independent of the true return 0. The AR(1) coefficient obtained fromregressing return series A on the one period lag of return sries B is O.~B = (0~L0a/T) (0~LOBL/T). The probability limit for the numerator is

plim(~[~L~a/T) �9 ^, ^ = phm(OLO/7),

and that for the denominator is

plim(O~LOBffT) = plim(O~OL/T) + plim(~l~L~lBL/7).

Since pl im(~ t ,~BL/7) r O, OaB is not a consistent estimator. One way to obtain a consistent estimator for the AR(1) coefficient is to partition the data

into two samples, estimate one index series for each sample, and then regress the change in one series (say, series A) on the lag change in the same series, using the lag change in the other series (series B) as an instrumental variable (IV). That is,

OAOBL (2)

The estimator in (2) is consistent, though like any IV estimator generally not unbiased. Like/3aB,/3re generates noise due to the data partition, so that different ways of splitting the data may produce different results.

In constructing real estate indexes using a repeat-sales Bayes estimator, Goetzmann (1992) formulated a model in which market indexes follow a random-walk behavior. In the next section, a more general AR(2) model is proposed. The proposed model can be used to construct price indexes, and also to estimate and test the serial correlation of market returns.

2. A Bayesian approach to the est imation of serial correlat ion and testing the random- walk hypothesis

Assume that the log price Yit of house unit i at time t is given by

t

Yit = o~i + Z Os + uit (i = 1, . . . , n; t = 0 . . . . , 7) (3) s=0

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where 0/i is the time-independent effect of house i on its sale price; 0~, representing the market return, is the change of the log index from time s - 1 to time s; and uit is an iden-

2 The data set con- ticaUy distributed normal noise term with mean zero and variance a u. tains the sales prices of n houses during a time span of T + 1 periods. Each house is sold at least twice during the time span, and there are N sales in all (N > 2n). Since no hedonic variable is included as a regressor, a single observation on the sales price for one house is not useful for estimation. Period 0 is treated as the base period, and its index is set to 1 (i.e., the log index is set to 0). Equation (3) can be rewritten in matrix form as

Y = Xa0/ + XoO + u, (4)

where 0/ = ( 0 / 1 , 0 1 2 . . . . , o/n) t 0 = (01 , 02 . . . . . OT) ', and Y is an N x 1 vector of sales prices. X a and X 0 are, respectively, N x n and N x T design matrices, with elements either zero or one corresponding to the house unit and the time period of the sales prices, Y.

For example, suppose that there are only three periods with observations on time 0, 1, and 2 sales, denoted as Yl0, Y12, Y21, and Y22. Then (4) becomes

Yl0 ] Yl2 Y21 Y22

1

1

0 0 ~ Ii ~

0 0/1 + 1 1 0/2 0 1 1

I01 + 1 I:::

n21 [_ 822

The market index, Os, is assumed to follow a stationary AR(2) process:

Os = ~P + OlOs-1 + P20s-2 + f:s (Io2l < 1, loll + 02 < 1), (5)

where es is an i.i.d, normal noise term and is independent o f uit for all s and t. Let 3'k denote the autocovariance of Os and Os-k. By solving the Yule-Walker Equation, it can be shown that

1 - P2 2 "/0 -- - - ~e

2 ')'1 = --O'E

'Ys = Ol'Ys-1 + P2'Ys-2

where

(s = 2, 3, 4, . . . ) (6)

g" = (1 + P2) (1 - P2 - - P l ) (1 - P2 + Pl).

To obtain the likelihood function of O1 and P2, it is convenient to deal with the trans- 2 There are n + T + 5 unknown 2 andz -- a2/a2 instead of o2 and a~. formed parameters o u

a and z, in this model. The posterior distribution of parameters, i.e., 0/, q~, 0, Pl, #2, o u, the unknown parameters conditional on the observed data, Y, is the product of the parameters' prior and the density function of the observations given the parameters. The parameters' prior


can be decomposed into the product of (i) the density function of the market return 0, conditional on the other parameters, and (ii) the prior density of the other parameters, namely,

p(ot, 0, ~b, 01, a2, tr], zIY')

2 z ) p ( Y I o t , O, 4~, 01, 02, au, p(ct, 0, ~b, /91, /92, O'u, 2 Z)

2 Z). o, p(~, ~, al, a2, ~2, z) p(ol~, ~, 01, 02, a2u, z) p(Yl~, o, ~,, 01, 02, Ou, (7)

Suppose that we have little prior information about the parameters other than 0, which is assumed to follow an AR(2) scheme, and that the parameters are a priori independent. We can assume flat priors for ct, q~, 01, 02, and z, and assume that the prior for ~ u 2 is pro- portional to 1/~ 2, i.e.,

p(ct, th, O1, P2, 02, Z) oc 1/g 2. (8)

Based on (6), we can write the conditional density function of the market returns, given the other parameters, as

p(OIo~, 4~ , p l , P2, 0"2, Z)

J - E I ~ 1 l a l - i /2 . 2.-r/2 1 1 - i r ~- a -1 o: tZau) e x p ] 2tr 2 z 1 - - P l - - P2

I ~ 1]) 1 - - P l - 0 2

(9)

where i r is the T • 1 vector of ones, and fl is the T x T matrix, with the element in the ith row, j t h column being ,Yli_jl/O2.

From (3), we can write the conditional density of the observations, given all the parameters, as

2 Z) p(YI~, O, 4~, m, / 9 2 , O ' u ,

oc (a2)-Ov/2) exp [(Y - X~t~ - X o O ) ' ( g - X~ot - XoO ) . 202

(10)

Substituting (8), (9), and (10), into (7), we can derive the conditional p.d.f, of the parameters (i.e., the posterior p.d.f, of the parameters), given the data:

2 zIY) oc (02)-(N/2)-11~[-1/2" 2.-T/2 p(o, , 0, dp, Pl, P2, au , ~za~) [Io lIo l e x p ; _ _ l _ l 1 _ iT (a ~ - 1 - ir 4a L 2a 2 z 1 - - P l - - P 2 1 - - P l - - jOE

q-5

( r - Xc~o, - X o O ) ' ( Y - X,~a - Xo0) I ~ .

JJ (11)

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Integrating or, tk, 0 2 and 0 out of (11) yields

P(Pl, P2, zlY) oc z-(T-1)/21•I-1/2 h-(N-n-1)/2(1 -- Pl -- /92)

I _ r 7r - 1/2 1 M~ fl-lM6 + XoMot.~ o (i ~fl- lir)l/2 z (12)

where

Mot = I N - Xot(X~Xot) -1 X~

M4 = I r - ir(i~f1-1 ir)- l f l -~

h = (Y - XoO)'Mot (Y - XoO) + 1 ~ , M ~ f l _ I M * M~fl- 'M~, + X , Mot Z

xduotxo

= (X M Xo) X Uotr

and Ir is the T • Tidentity matrix. Note that (12) is a proper density function (see Ap- pendix 1 for proof), and thus can be integrated with respect to Pl, P2, and z. The posterior density of the AR(2) coefficients Pl and P2 can be obtained by numerically integrating z out of (12), and then dividing by a normalizing factor.

Let the right-hand side of (12) be denoted as g(Pl, P2, z); then the posterior density of" t91 and P2 can be computed by

P(P l , P2[ Y) = g(Pl, /)2, z )dz (14)

f l l r l - o 2 r = Jp2-1 J o g(Pl, P2, Z) dzdpldp2

Given the posterior density of (Pl, P2), we can compute the posterior means of Pi, P2, and the posterior mean and density of 3'k/3'0, the t-period-lag serial correlation of the market returns.

Because of infrequent trading in the real estate market, the market indexes are unobserv- able. Consequently, the two-step methods have an inherent shortcoming; the first-step estimated indexes are treated as true indexes in the second step. The advantage of the Baye- sian approach is that the unknown true indexes are treated as random variables, and the data set need not be arbitrarily partitioned.

The change of variable technique is not applicable for deriving the posterior density of the serial correlation, 3'k/'Y0 because the ~[k/~/o terms in (7) are not one-to-one functions of ,o I or /)2- By the distribution function technique, we can compute

Pr(b i <_ (~/k/~/o) <_ bi) = f f X[bi,b fl ('yk/'yo)p(pl, p2lY)dpldP2, (15)


where X is the indicator function. Dividing (15) by (bj - bi) gives the average density of 3'k/y 0 in the interval [bi, bj]. There is no analytical solution to (15), but the density function can be well approximated by choosing an interval as small as desired.

In (14), X 0 and M~, are matrices of size N x n and N x N, respectively. In the data set used here, N is as^large as 31,160. Some technical details for calculating XdM~X 0, XdM~,Y, and (Y - XoO)'M,~(Y - XoO ) are discussed in Appendix 2.

The AR(2) model presented in this section reduces to the random-walk model when Pl = P2 = 0. In order to test the random-walk hypothesis in the real estate market, we can compute the (1 - ix) highest posterior density interval (HPDI) for serial correlation and examine whether zero is inside the HPDI. If the HPDI does not include zero, then we would reject the random-walk hypothesis at the 1 - et level.

An alternative way to test the random-walk hypothesis is to compute the posterior odds ratio in favor of the random-walk model relative to the AR(2) model. We would accept the random-walk model, if the posterior odds ratio is greater than one, and the AR(2) model if the posterior odds ratio is less than one. Denote the denominator of (14) as 12, and the numerator of (14) valued at Pl = P2 = 0 as I 0. The posterior odds ratio in favor of the random-walk model is K02 = lo/(0.25 * I2). The scaler, 0.25, comes from the flat prior for (Pl, P2).

2 For inference on the constant term ~, we integrate (8) with respect to or, 0 and a u to derive the joint likelihood function of ~, Pl, P2, and z,

p(t#, Pl, P2, z]Y) or z-T/2[~ I-1/2 lz ~-1 + X~MtxX 0 1/2

[ f + (~b - ~)2 i~.~bir/(1 - P l - p 2 ) 2 1 - ( N - n ) ~ 2 (16)

where

I Z 1- Me~Xol - 1 ' 1 fl-1 •-1 + X~ X o M ~ X o r z

f = ( Y - X o O ) ' M ~ ( Y - XoO) + ~0 - iT t._

= (1 - Pl -- P2)(iTdgiT) -1 i}~bO.

l _ p~ _ p2 ~ ~0 - iT 1 - P l - / 9 2

Equation (16) indicates that the conditional distribution of th, given Pl, P2, and z, is a t-distribution with degree of freedom N - n - 2, mean ~, and variance a ~, where

a~ = (1 - Pl - p2)2(iTr - l f / ( N -- n -- 4). (17)

The posterior t-distribution of ff given Pl,/92, and z has many degrees of freedom, since there are thousands of houses in the data set. Therefore, it is well approximated by a normal distribution. Its marginal posterior density obtained by numerically integrating Pl, P2, and z out of (12) would still be close to a normal distribution, so only mean and variance are needed for inference.

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3. Empirical results

In this section, an application of the foregoing methodology is discussed. The data set used is the one used by Case and Shiller (1987, 1989). It contains quarterly sales prices of single- family homes in four metropolitan areas, Atlanta, Chicago, Dallas, and San Francisco, from 1971 to 1986. Each house included in the data set was sold twice during this period and had no apparent quality change between sales. The number of these double-sale houses in Atlanta, Chicago, Dallas, and San Francisco was 8945, 15530, 6669, and 8066, respectively.

Alternative estimates of the AR(2) parameters are computed by using two-step methods and a Bayesian method. The two-step methods estimate price indexes in the first step, and then estimate AR(2) coefficients in the second. Three different two-step estimators are computed. The first (two-step, one-sample estimator--2S1S) is produced by regressing the change in the estimated log index on the lagged change of the same index series, where the indexes are calculated using the BMN OLS method with the whole data set. The second two-step estimation method (two-step, two-sample OLS estimator--2S2S-OLS), following Case and Shiller (1989), randomly partitions the data into two samples, calculates the BMN OLS indexes for each sample, and then regresses the change in log index of one series on the lagged change of the other series. The third method (two-step, two-sample instrumental variable estimator--2S2S-IV) randomly splits the data into two parts, estimates two index series, and then regresses the change in one index series on the lagged change in the same series, using the lagged change of the other series as an instrumental variable. The Baye- sian estimator is computed according to the Bayesian model presented in section 2. For two-step estimators, the OLS estimates of the AR(2) coefficients, their 95 % confidence intervals, and the implied autocorrelation function (ACF) are computed; while, for the Baye- sian model, the posterior mean, the 95 % HPDI, the implied ACFs, the posterior densities of the AR2 parameters, and the posterior densities of the ACFs are computed. The implied ACFs are derived by substituting the posterior means of Pl and P2 reported in Table (1) into (6). Both nominal sales price and real sales price (log nominal sales price minus log citywide consumer price index) are used as the dependent variable, Y.

Table 1 displays the point and interval estimates for the AR(2) parameters in each city. It shows that there are noticeable discrepancies among the estimates for alternative methodologies. For Atlanta, each of the five two-step estimates of Pl is significantly negative at the 95% or the 90% level, while the Bayesian estimates are insignificant. For San Francisco, the Bayesian estimates of Pl are significantly positive at the 95% level; in contrast, the 2S1S estimates are insignificantly negative, while the 2S2s estimates yield conflicting results. The 2S2S-OLSAB estimates (obtained by regressing the market return computed from index series A on the return computed from index series B) are strongly positive, but the 2S2S-OLSBA estimates are insignificantly negative. The 2S2S-IV A estimates (obtained by regressing the market return computed from index series A on its lag return, using the return computed from index series B as an instrumental variable) of P2 are significantly negative, while the 2S2S-IV B estimates of P2 are not. The two 2S2S- OLS estimates of P2 for San Francisco are both significant, but one is positive, the other is negative. For Dallas, the Bayesian posterior mean of P2 is positive although insignificant, but the 2S1S and the 2S2S-OLSBA estimates of P2 are strongly negative. In general, the Bayesian estimates tend to be more positive than the 2S1S and 2S2S-OLS estimates.


Table 1. Estimates of AR(2) parameters with alternative models.

Nominal Index Real Index

Point Estimate (95 % Interval) Point Estimate (95 % Interval)

Atlanta

2S1S ol -0.397 (-0.639, -0.154) -0.432 (-0.680, -0.184)

02 -0.019 (-0.262, 0.224) 0.014 (-0.236, 0.265)

2S2S-OLSAB Pl 4).178 (-0.371, 0.014) -0.190 (-0.381, 0.001)

02 0.174(-0.019, 0.366) 0.214 (0.021, 0.406)

2S2S-OLSBA Pl -0.369 (-0.689, -0.049) -0.421 (-0.750, -0.092)

P2 -0.083 (-0.404, 0.238) -0.056 (-0.389, 0.277)

2S2S-IVA ol -0.362 (-0.721, 0.003) -0.370 (4).738, -0.002)

P2 0.295 (-0.063, 0.653) 0.354 (-0.015, 0.723)

2S2S-IVB Pl -0.494 (-1.011, 0.022) -0.534 (-1.060, -0.008)

P2 4).151 (-0.678, 0.376) -0.129 (-0.674, 0.416)

BE 01 0.083 (-0.351, 0.517) -0.140 (-0.489, 0.276)

P2 0.169 (4).254, 0.579) 0.197 (-0.142, 0.579)

Chicago

2S1S ol 0.179 (-0.078, 0.435) 0.166 (-0.093, 0.426)

P2 0.081 (-0.168, 0.330) 0.105 (-0.150, 0.359)

2S2S-OLSA~ Pl 0.139 (-0.095, 0.372) 0.105 (-0.108, 0.319)

P2 0.209 (-0.021, 0.439) 0.207 (-0.005, 0.419)

2S2S-OLSnA Pl 0.370 (0.120, 0.620) 0.397 (0.114, 0.679)

o2 0.037 (-0.207, 0.281) 0.018 (-0.260 0.295)

2S2S-1VA Pl -0.349 (-1.707, 1.008) 4).100 (-0.648, 0.448)

,o 2 0.674 (-1.116, 2.464) 0.470 (-0.154, 1.095)

2S2S-IVB 01 1.473 (-2.287, 5.234) 0.776 (-0.345, 1.897)

P2 -1.392 (-3.885, 1.099) -0.446 (-1.340, 0.449)

BE Pl 0.425 (-0.008, 0.850) 0.354 (-0.008, 0.714)

P2 0.204 (-0.193, 0.643) 0.216 (-0.141, 0.552)

Dallas

2S1S Pl 4).019 (-0.266, 0.227) -0.162 (-0.399, 0.076)

P2 -0.272 (-0.486, -0.058) -0.313 (-0.518, -0.108)

2S2S-OLSAB 01 0.109 (-0.124, 0.342) 0.004 (-0.213, 0.220)

P2 0.001 (-0.233, 0.234) -0.052 (-0.269, 0.165)

2S2S-OLSBA ol 0.292 (0.039, 0.545) 0.270 (--0.008, 0.547)

O2 -0.286 (-0.484, -0.088) -0.306 (-0.517, -0.095)

2S2S-IVA Pl 0.478 (-0.697, 1.652) 0.139 (-0.539, 0.818)

o2 -0.238 (-1.597, 1.120) -0.199 (-1.080, 0.683)

2S2S-IVB 01 1.795 (-2.114, 5.705) 1.034 (-1.831, 3.900)

02 -2.144 (-5.827, 1.539) -1.756 (-4.164, 0.652)

BE Pl 0.396 (-0.137, 1.135) 0.295 (-0.207, 0.824)

P2 0.335 (-0.291, 0.841) 0.184 (-0.291, 0.684)

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Table 1. Continued.


Point Estimate (95% Interval) Point Estimate (95% Interval)

San Francisco

2S1S /91 -0.070(-0.327, 0.186) -0.095 (-0.350, 0.160)

P2 0.162 (-0.091, 0.414) 0.174 (-0.079, 0.426) 2S2S-OLSAa Pl 0.428 (0.202, 0.653) 0.399 (0.175, 0.623)

P2 -0.259 (-0.479, -0.040) -0.217 (-0.436, 0002) 2S2S-OLSBA Pl -0.102 (-0.300, 0.096) -0.077 (-0.288, 0.135)

P2 0.610 (0.411, 0.809) 0.615 (0.401, 0.828)

2S2S'IVA Pl -0.152 (-2.512, 2.208) -0.401 (-3.083, 2.281) P2 -2.094 (-3.089, -1.099) -2.273 (-3.343, -1.204)

2S2S-IV B ,ol -2.610 (-3.662, -1.557) -2.884 (-4.124, -1.644)

Pz 0.157(-2.309, 2.622) -0.121 (-3.245, 3.003) BE Pl 0.479 (0.084, 0.936) 0.598 (0.084, 1.146)

02 0.147 (-0.262, 0.525) -0.015 (-0.476, 0.425)

Note: 2S1S denotes the two-step, full-data estimate. 2S2-OLSAB denotes the two-step, two-sample estimates obtained by regressing the sample A index on the sample B index. 2S2S-IV A denotes the two-step, two-sample estimates obtained by regressing sample A returns on the lag returns of sample A, using the lag returns from sample B as an instnunental variable. BE denotes the Bayesian estimate. For the two-step estimate, the point. estimate is the OLS estimate and the 95% interval is the 95% confidence interval. For the Bayesian estimate, the point estimate is the mean of the posterior distribution, and the 95% interval is the 95% highest posterior density interval.

Most of the two-step IV estimates have much wider 95% confidence intervals than the estimates obtained from other methods. Half of the two-step IV estimates are outside the stationary region, while the estimates obtained from other methods are all within the stationary region (i.e., I p21 < 1, I pll + p2 < 1). The bias and noise due to the data partition, especially when there is low correlation among the partitioned series, account for the high standard error of the two-step IV estimates. When the correlation between the return as independent variable and the return as instrumental variable is low, the standard error of the estimated serial correlation tends to be large. To make the two-step instrumental variable estimator useful, one has to randomly partition the data into two samples and get highly correlated return series. But even so, the estimated serial correlation still varies, depending on how the data are partitioned.

Table 2 reports the implied ACFs for lags of one to four quarters. The implied ACFs of one to four lags derived from the Bayesian estimates are all positive except for the Atlanta real index case; those derived from the 2S1S and 2S2S-OLS estimates, however, are all positive for Chicago only. The one-quarter-displaced autocorrelations derived from the Baye- sian estimates are more positive than those derived from the two-step estimates for each of the four cities. In general, the autocorrelations derived from the Bayesian estimates tend to be positive and greater than those derived from the two-step estimates.


Table 2. Implied autocorrelation of quarterly returns.

Nominal Index Model Real lndexModel

~1 ~2 63 64 61 62 ~3 ~4

Atlanta 2S1S -0.39 0.14 -0.05 0.002 -0.44 0.20 -0.09 0.04 2S2S

OLSAa -0.22 0.21 -0.08 0.05 -0.24 0.26 -0.10 0.07 OLSBA -0.34 0.04 0.01 -0.01 -0.40 0.11 -0.02 0.004 IV A -0.51 0.48 -0.33 0.26 -0.57 0.57 -0.41 0.35 IV B -0.43 0.06 0.03 -0.03 -0.47 0.12 0.00 -0.01

BE 0.10 0.18 0.03 0.03 -0.17 0.22 -0.07 0.05

Chicago 2S1S 0.20 0.12 0.04 0.02 0.19 0.14 0.04 0.02

2S2S OLSAB 0.18 0.23 0.07 0.06 0.13 0.22 0.05 0.05 OLSBA 0.38 0.18 0.08 0.04 0.40 0.18 0.08 0.03 IV a . . . . . 0.19 0.49 -0.14 0.24 1V B . . . . 0.54 -0.03 -0.26 -0.19

BE 0.53 0.43 0.29 0.21 0.45 0.38 0.23 0.16

Dallas 2S1S -0.02 -0.27 0.01 0.07 -0.12 -0.29 0.09 0.08 2S2S

OLSAB 0.11 0.01 0.002 0.002 0.004 -0.05 -0.0004 0.003 OLSBA 0.23 -0.22 -0.13 0.03 0.21 -0.25 -0.13 0.04 IV A 0.39 -0.06 -0.12 -0.04 0.12 -0.18 -0.05 0.03 IV B . . . . . . . .

BE 0.60 0.57 0.43 0.36 0.36 0.29 0.15 0.10

San Francisco 2S1S -0.08 0.17 -0.03 0.03 -0.12 0.18 -0.04 0.04 2S2S

OLSAa 0.34 -0.11 -0.14 -0.03 0.33 -0.09 -0.11 -0.02 OLSBA -0.26 0.64 -0.22 0.41 -0.20 0.63 -0.17 0.40 IV A . . . . . . . . IV B . . . . . . . .

BE 0.56 0.42 0.28 0.20 0.59 0.34 0.19 0.11

Note: 6 t is the implied t-quarter-lagged autocorrelation of market returns computed based on the estimates in Table 1. For the two-step methods, the autocorrelation is computed only when the estimated return process is stationary.

Table 2 shows the e r r o r s i n t r o d u c e d by pa r t i t i on ing the da ta set into two samples to be

s ignif icant . T h e d i s c r epanc i e s b e t w e e n the two 2 S 2 S - O L S es t imates o r be tween the two

2S2S- IV es t ima tes for i m p l i e d au toco r re l a t i on a re qui te big. For example , in San F ran -

cisco, the two 2 S 2 S - O L S es t imates for au toco r r e l a t i on wi th o n e - p e r i o d lag a re of d i f fe rent

s ign (0 .34 a n d - 0 . 2 6 ) , as a re the es t imates w i th two-pe r iod lag ( - 0 . 1 1 and 0 .64) . For

Chicago , in the real i n d e x case, the two 2 S 2 S - I V es t imates o f au toco r r e l a t i on wi th one-

pe r iod , two-per iod , or t h r ee - pe r i od lag are o f d i f fe ren t s ign as well . O n the o the r h a n d ,

the Bayes ian es t ima to r p rov ides an accu ra t e m e a s u r e of the a u t o c o r r e l a t i o n in rea l estate m a r k e t re tu rns .

150 KUO

In the model described in section 2, Pl is restricted to lie between - 2 and 2, and P2 between -1 and 1. The marginal posterior densities of both parameters are plotted in Figures 1-8, and the 90% and 95 % highest posterior density contours (HPDC) are plotted in Figures 9-16. The Bayesian estimates of Pl for the nominal index are significantly positive for Chicago, Dallas, and San Francisco, but none of the p2s or the joint distributions of (Pl, P2) are significantly positive. It is interesting that the joint density for the Dallas nominal index model has two peaks (see Figures 5 and 13).

These posterior densities are not highly concentrated. The shortest 95% HPDI for p~ spreads over an interval of length 0.72, and that for P2, over 0.69. From Table 1, it can be observed that the 95 % HPDI of the Bayesian estimate is larger than the 95 % confidence interval of the 2S1S estimates. The relatively larger variance of the Bayesian estimator reflects the fact that the Bayesian model treats the indexes as random variables, while the smaller variances of the 2S1S estimators reflect the fact that the estimated indexes from the first step are treated as the true index in the second step.

Given the posterior joint density of (Pl, P2), the posterior density of the autocorrelation can be computed. The posterior mean and the 95 % HPDI of autocorrelation functions of quarterly and yearly returns (ACFq and ACFy) are reported in Table 3 and Table 4 for lags of one to five periods, and plotted in Figures 17-32 for lags of one to 20. Note that the implied one-period-lagged autocorrelation for the Bayesian estimate in Table 2 is slightly different from the autocorrelation in Table 3, because the former is computed assuming that the joint density of (Pl, P2) is degenerate and concentrated at the posterior mean of

(pl, p2). The Bayesian estimates of the ACFs suggest the existence of strong positive serial cor-

relation in the single-family housing market. The one-quarter-lagged and two-quarter-lagged autocorrelations are strongly positive for Chicago and San Francisco for both nominal and real index models, and, in the case of Dallas, for the nominal index model. The lag-one- year ACF is strongly positive for Dallas and San Francisco for the nominal index model, and, in the case of Chicago, for both the nominal and real index models.

The posterior odds ratio in favor of the random-walk model, K02, in the nominal index case for Atlanta, Chicago, Dallas, and San Francisco is 2.669, 0.025, 0.007, and 0.010, respectively; in the real index case, the posterior odds ratio is 2.532, 0.052, 0.778, and 0.015, respectively. According to the posterior odds ratio criteria, the data support the AR(2) model over the random-walk model for Chicago, Dallas, and San Francisco.

From (12), the posterior density of z can be obtained by numerically integrating out 2 2 (Pl, 02). Recall that z -- a~/au is the ratio of the variance of market index error to that

of the individual house price error. The posterior means ofz for Atlanta, Chicago, Dallas, and San Francisco are respectively 0.025, 0.013, 0.011, and 0.030 in the nominal index model, and 0.039, 0.024, 0.017, 0.024 in the real index model. The posterior means of z suggest that the standard deviation of individual house error is five to ten times larger than that of the market index error.

Table 5 reports the mean and the scaled mean (defined as mean divided by standard deviation, which corresponds to the t-value in classical sampling inference) of the posterior distribution of the log quarterly trend, ~b. Although the quarterly trend of the nominal housing price model is strongly positive for Atlanta, Chicago, and San Francisco, none of the quarterly trends of the real housing price model are statistically significant. Among the four


Fig. l: Marginal Density of p~ for Atlanta

0 [ ...... nominal index model �9 ~ I t ~,,, ~ _ real lnd,x model

"E \

q ~ 2 . D - 1 . 6 - 1 . 2 - - 0 . 8 - - 0 , 4 - 0 . 0 0.4 0.8 1.2 1.6 2.0

Pl

Fig. 3: Marginal Density of Pl for Chicago

..... nominol index model / ' ~ real index model

~ 2 . 0 - 1 . 6 - 1 . 2 - 0 . 8 - 0 . 4 - 0 . 0 0.4 0.8 1.2 1.6 2.0

Pl

Fig. 5: Marginal Density of p~ for D0flas

ol

c~

~o o ~ . a- d

_ o ..... nominel index model real i ndex m o d e l

-o

,o ~

O,r o- d

o . . . . . ,

Q 2 . 0 - 1 . 6 - 1 . 2 - 0 . 8 - 0 . 4 - 0 . 0 0 .4 0 .8 1.2 1.6 2.0

Pl

Fig. 7: Marginal Density of Pl for S.F,

._(2 ~

~o 0,r

o r 0.4 0.8 1.2

Pl

nominal index model - - real index model

/ \

1 . 6 2.0

Fig. 2: Marginal Density of P2 for Chicago

- - ' nominal index model . - - - rea l i ndex m o d e l

o '

i

(

H . O - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 - 0 , 0 0.2 0,4 Q.6 0.8 1.0

P2

Fig. 4: Marginal Density of P2 for Chicago

. . . . . : ~ o t [ ----~" nominal index mod"--~el .~11--_ :eo~ ~2d . . . . de__.____~

O ~ ,, ~ ,! "~ O //

~'d /"

c } 1 . 0 - 0 . 8 - 0 . S - 0 . 4 - 0 . 2 - 0 . 0 0.2 04 0.6 0.8 1.0

P2

Fig. 6: Marginal Density of P2 for Dallas -k e~ . . . . . . . . .

Dare,hal nd0x roe' l �9 ~ ..- ,,..

0 " 2 ~\ "E =0 '\

o -e '~ o- 6 ',,

o , i ';':- ~ 1 . 0 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 - 0 . 0 0.2 0.4 Q.6 0.8 1.0

P2

Fig. 8: Marginal Density of P2 for S.F.

.[ r i - - nominal index model

~ , i J [ - - reel index model

~N r ,2 /"

'E "

N o /- o; /

q 01.0 -0 .8 -0 .6 - 0 . 4 -0 .2 -0 .0 0.2 0.4 0.6 0.8 1.0

/O2

Figures 1-& Marginal posterior densities of the autoregressive parameters of the AR(2) market return process.

152 KUO

ff l

q

Figure 9: HPDC for Attanto Nominal Return o

90Z HPDC

-P2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1,5 2.0

Pl

Figure 10: HPDC for Atlanta Real Return

' ~ -90~ HPDC 95~ HPDC

r

2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Pl

Fig. 11: HPDC for Chicago Nominal Return Figure 12: HPDC for Chicago Real Return o

,5

u3

q

[-- 907' .Po~l 957. HPDCJ

-1"2,0 -1,5 -1.0 -0.5 0,0 0.5 1,0 1,5 2.0

~01

Figure 13: HPDC for Dallas Nominal Return o

" " 90~ HPOC 95~ HPDC

q

-t2.0 -1.5 -1,0 -0.5 0.0 0.5 1.0 1.5 2.0 Pl

Figure 15:HPDC for S.F. Nominal Return

90Z HPDC I 95z HPOCl

- - 9 0 7 , HPDC 957. HPDC

.2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 115 " 2.0

Pl

Figure 14: HPDC for Dallas Real Return

~ gOZ HPDC 957, HPDC

:.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 Pl

Figure 16: HPDC for S.F. Real Return

--- 907. HPOC 957, HPDC

.-t2.0 -1.5 -1.0 -0,5 0.0 0.5 1.0 1.5 2.0 -t2,0 -1,5 -1.0 -0,5 0.0 0.5 1.0 1.5 2.0

Pl Pl

F/gums 9-I 6. Highest posterior density contours of the autoregressive pmmneters of the AR(2) market return process.


o~ o

o,6

0 o

~0o .o~ o

~ d L) 0

O 0

Figure 17: ACFq for At lanta Nominal Index

,~. posterior mean I 95Z lower bound 957. upper bound

~

1"1 /

lagged quarters

Figure 19: ACFq for Chlcago Nominal Index

~ posterior mean J ,.. 957. lower bound J "t ,. 957. upper boundJ

.'~"".. ~ ~ - . : ' -_*"T"

legged quarters

Figure 21: ACFq for Dallas Nominal Index 04 .c

.s o

o=o

,s o

o 0

*-. I ~'- posterior mean t "~--.,.. "" 957' lower bound I

"*",,......,, *'-- 957' upper bound I

s _ _ = -e----. 4' ~,.,. " % - _

v , / - . ....... ...-

�9 ~ " ~ " ~ ~ ~0 ~~ <4-"~~ 18 ~0 lagged quarters

Figure 23: ACFq for S.F. Nominal Index

posterior mean �9 ,~, 95% lower bound

',, 957. upper bound

�9 ~"~" .... * " ~ "~'T"~-*_~. ,*---

�9 ~ ' ~, ' & ' 8 ' < 0 ? ~ ' ~ ' ~ ? ~ ~ ' e =0 logged quarters

Figure 18: ACFq for At lanta Reat Index

p o s t e r i o r m e a n ~ I ~''" g57' l o w e r b o u n d I

. - /~ I~ ~ \ ,

/ \ ~( / ' , / ~, . ~ - - . . ~ - ~ " ~ r

t 2 4- 6 8 10 12 14 16 18 20

lagged quarters

Figure 20: ACFq for Chicago Real Index

~l . . . . . . . . .

":I [ posterior mean c 00 , / * - ' - ~ 957. l owe r b o u n d

~ o. - . . T - ' T " ~ . " - ~ _ . . - - ~ . * -

"L. . . . . . . . . '-~/ , "u . . . . . . . . O0 2 4 6 8 10 12 14 16 18 20

~o

o=o

legged quarters

Figure 22: ACFq for Dallas Reel Index

t--00 .o<5 o

~ d

" ~ o

lagged quarters

*, ~-- p o s t e r i o r m e a n I " , , . ** - 957, l owe r b o u n d

'~ - 957. u p p e r b o u n d

" ~ " e ' ~ - - - * ~ - . e ..

.# " 5 . . . . i i x ~ f / i , i i i , i , i i

t a 2 4, 6 8 10 12 14 16 18 . 20

lagged quarters

Figure 24: ACFq for S.F. Real Index ~f

~_. posterior mean I ... 95z lower bound

"..�9 957' upper bound

20

Figures 17-24. Posterior means and 95% highest posterior density intervals of the autocorrelation of quarterly market returns for lag 1-20 quarters.

154 KUO

Figure 25: ACFy for Atlanta Nominal Index e~

coo I ~-" 957. lower bound } ,~ 1.-- ~s~. upper ~a0n~ 1 o

Figure 26: ACFy for Atlanta Real Index

(J o "Sq O0

t

l * ' - poster lor meon I c=~ f*"" 95Z lower bound J

"-" gSZ upper bound I

~ " . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y

"C.

lagged years lagged years

Figure 27: ACFy for Chicago Nominal index Figure 26: ACFy for Chicago Real Index r ~1

' "' Coo g5Z lower bound ~ c ~ �9 0 o 95Z upper boundJ o c

1'o " ~ " i ; ~ " ~'o ' ~'~ ' i '~ " i ~ ~ ' ~o ~ . . . . . . . . . . . . . . . . . . . . . . I 2 4 6 8 10 12 14 1~ 18 20 lagged years

Figure 2g: ACFy for Dallas Nominal Index

" : - . : ~oet; ; ior r~eoo ] coo . /"" 95Z lower boundl :~

o %. _o

-I-'1:3

~L "~ I0 2 4 6 8 10 12 14 16 18 20

l agged years

Figure 51: ACFy for S.F. Nominal Index

I '~-- posterior mean J coo ]* ' - 95Z lower bound

,t I

logged years

lagged years

Figure 30: ACFy for Deltas Real Index

. . . . . . . . . . .-poet;r,or m'.on f l ~"- gSZ lower bound J |

", J ..... gsz upper boundJ ]

1

I lagged years

Figure .32: ACFy for S.F. Reel fndex ~N

l ~i~----7posto.or mean c a | [ " ' - 957, lower bound

"-0~5I I .... 95Z upper bound

0 x

:1 lagged years

Figures 25-32. Posterior means and 95 % highest posterior density intervals of the autocorrelation of annual market returns for lag 1-20 quarters.


Table 3. Autocorrelation of quarterly returns.

Nominal Index

Mean 95% HPDI Mean

Atlanta ~ql 0.080 (-0.419, 0.579) -0.180 ~5q2 0.226 (-0.169, 0.643) 0.258 ~q3 0.008 (-0.253, 0.352) -0.085 6q4 0.096 (-0.114, 0.330) 0.111 ~q5 0.005 (-0.154, 0.226) -0.047

Chicago 6ql 0.517 (0.097, 0.857) 0.443 ~q2 0.459 (0.130, 0.806) 0.403 t$q3 0.292 (-0.037, 0.676) 0.238 /~q4 0.252 (-0.038, 0.593) 0.203 /~q5 0.179 (-0.055, 0.523) 0.139

Dallas tSql 0.536 (0.011, 1.000) 0.329 ~q2 0.629 (0.310, 0.933) 0.343 t$q3 0.396 (0.012, 0.875) 0.142 ~q4 0.432 (0.105, 0.854) 0.166 ~q5 0.294 (-0.012, 0.796) 0.082

San Francisco t~ql 0.546 (0.168, 0.870) 0.571 62 0.439 (0.141, 0.837) 0.372 33 0.284 (-0.012, 0.667) 0.207 iiq4 0.227 (-0.048, 0.567) 0.138 /~q5 0.164 (-0.056, 0.471) 0.090

6qt is the t-quarter-lagged autocorrelation of market returns.

Table 4. Autocorrelafions of annual returns.

Real Index

95% HPDI

(-0.621, 0.260) (-0.054, 0.843) (-0.390, 0.207) (-0.070, 0.376) (-0.247, O. 162)

(0.083, 0.841) (0.069, 0.754)

(-0.060, 0.586) (-0.038, 0.506) (-0.064, 0.436)

(-0.172, 1.000) (-0.073, 0.781) (-0.234, 0.576) (-0.135, 0.517) (-0.202, 0.394)

(0.206, 0,867) (0.033, 0.711)

(-0.104, 0.587) (-0.146, 0.485) (-0.106, 0.431)

Nominal Index

Mean 95% HPDI Mean

Atlanta /~yl 0.140 (-0.155, 0.435) 0.070 /iy 2 0.027 (-0.084, 0.143) 0.010 ~y3 0.008 (-0.051, 0.085) 0.002 tSy 4 0.003 (-0.032, 0.042) 0.001 ~y5 0.002 (-0.023, 0.036) -0.0003

Chicago tSy I 0.405 (0.034, 0.705) 0.356 t~y 2 0.149 (0.124, 0.437) 0.113 6y 3 0.069 (-0.114, 0.283) 0.047 ~y4 0.037 (-0.091, 0.187) 0.023 ~y5 0.021 (-0.065, 0.149) 0.013

Real Index

95% HPDI

(-0.169, 0.309) (-0.044, O. 110) (-0.031, 0.034) (-0.013, 0.020) (-0.012, 0.014)

(0.045, 0.700) (0.131, 0.370)

(-0.100, 0.219) (-0.070, O. 140) (-0.053, 0.110)

156 KUO

Table 4. Continued.


Mean 95% HPDI Mean 95% HPDI

Dallas ~yl 0.559 (0.232, 0.850) 0.274 (-0.077, 0.760) ~Sy 2 0.296 (0.088, 0.661) 0.083 (-0.141, 0.319) ~y3 0.176 (-0.092, 0.511) 0.035 (-0.109, 0.202) ~y4 0.114 (0.127, 0.425) 0.018 (-0.079, 0.130) ~y5 0.078 (-0.091, 0.354) 0.010 (-0.071, 0.099)

San Francisco ~yl 0.382 (0.030, 0.684) 0.283 (-0.077, 0.575) by2 0.126 (-0.107, 0.438) 0.066 (-0.133, 0.264) ~y3 0.054 (-0.094, 0.247) 0.025 (-0.077, 0.150) ~Sy 4 0.027 (-0.079, 0.158) 0.011 (-0.056, 0.094) ~y5 0.015 (-0.056, 0.114) 0.006 (-0.035, 0.076)

6yt is the t-year-lagged autocorrelation of market returns.

Table 5. Estimates of log quarterly trend.

Nominal Index Model Real Index Model

Mean Scaled Mean Mean Scaled Mean

Atlanta 0.0124 2.918 -0.0001 -0.027 Chicago 0.0063 2.157 0.0005 0.266 Dallas 0.0053 1.659 0.0024 1.118 San Francisco 0.0094 2.201 0.0036 1.506

cities, Atlanta has the highest nominal quarterly trend and the lowest real quarterly trend. The strong nominal quarterly trend and insignificant real trend suggest that real estate is a good cross-hedge for inflation.

4. Seasonality

In an efficient market, one would expect abnormal returns of financial assets due to seasonality to disappear. Case and Shiller (1987) reported in their appendix that there is not enough evidence to support the existence of seasonal abnormal returns. By regressing the change in the WRS log nominal index on its lag and seasonal dummies, they found that the seasonal dummies are insignificant except for Chicago. In this section, the foregoing Bayesian model is modified to incorporate seasonal dummy variables. Let rj be the constant rate of return associated with the j th quarter. Equation (5) is now modified to become:

4

Os = PlOs -1 + P20s-2 + E ~ j V j + 6.s, j= l

(18)


where Dj is the dummy for the j th quarter and )~j = log(1 + rj). Using the same pro- cedure described in section 2, we may obtain the posterior distributions of the AR(2) coefficients and the dummy coefficients. Like the quarterly trend, th, in section 2, the posterior distribution of ),j's can be well approximated by a normal distribution; thus, we need only compute their posterior means and variance-covariance matrix to do inference.

To test the hypothesis that there are no overall differences among the seasonal dummy coefficients (i.e.,)~1 = )~2 = X3 = )~4~), denote the means and variance-covariance matrix of the seasonal constant terms 9~ as X and Ex, respectively. Let

[ 121 i 1 21 [1 1001 d = d23 = ~2 - ~k3 and H = 0 1 -1 0 . d34 ~k3 -- h4 0 0 1 -1

Premultiplying by H, we find that the linearly transformed X is roughly distributed as a normal distribution,

H ~ - d - N ( H ~ , H ~ x H ' ) .

If the point [0, 0, 0]' lies outside the (1 - ~) H.ED. region of the r.v. d, we can say that the possibility of (hi -- h2 = )k3 ~- ~k4) is lower than a. The density of the normal distribution N(/-/h,/T2xH' ) is a monotonically decreasing function of the quadratic form (d - / ~ ) '(/-/~xH') -1 (d - Hh). Therefore, the condition that [0, 0, 0] ' lies outside the ( 1 - o0 H.ED. region of d is equivalent to

~2 ~ (/_/~),(/_/~xH,)-I (/_/~) > X2(3), (19)

where x~(r) denotes the constant for which

P r ( w > = x~(r) ) = ot

where w has a X 2 distribution with r degrees of freedom. Similarly, we can also test the hypothesis of pairwise equality of the seasonal dummy coefficients and the hypothesis that the seasonal dummy coefficients are jointly zero.

Table 6 reports the posterior means and the 95 % HPDI of the AR(2) coefficients, and the means and scaled means of the Xjs for the seasonality model. The inclusion of the seasonal dummies tends to lower the significance of the autoregressive coefficient, but the positive significance of the ACFs still exists for Chicago, Dallas, and San Francisco. As for the seasonal dummy coefficients, none are statistically significant in the real index case, but the first three quarters for Chicago and San Francisco exhibit significantly positive effects in the nominal index case. The returns due to the seasonal dummies tend to be strongest in the second quarter, followed by the first quarter, then the third quarter, with the fourth quarter being the smallest. Table 6 suggests that there is relatively strong demand for residential houses in the first half of the year, and relatively slack demand in the fourth quarter.

158 KUO

Table 6 Estimates for AR(2) parameters and seasonal dummy coefficients.

Nominal Index Model Real Index Model

Scaled Mean Scaled Mean Mean or 95 % HPDI Mean or 95 % HPDI

Atlanta ol -0.052 (-0.489, 0.437) -0.263 (-0.616, 0.113) 02 0.180 (-0.254, 0.579) 0.116 (-0.254, 0.480) X 1 0.0173 1.9112 0.0019 0.1525 )~2 0.0210 2.1652 0.0045 0.3566 9~3 0.0112 1.3488 -0.0043 -0.3360 )'4 0.0081 0.9795 -0.0032 -0.2520

Chicago Ol 0.298 (-0.211, 0.783) 0.366 (-0.063, 0.783) 02 0.285 (-0.294, 0.761) 0.228 (-0.141, 0.599) )k I 0.0097 2.A.A.n. n. 0.0046 1.6098 X 2 0.0103 2.5008 0.0029 -1.1151 X 3 0.0068 2.0293 -0.0013 -0.5247 •4 0.0011 0.4191 -0.0045 -1.4999

Dallas O1 0.180 (-0.273, 0.660) 0.102 (-0.395, 0.577) 02 0.361 (-0.113, 0.807) 0.109 (-0.348, 0.588) ~k 1 0.0110 1.6196 0.0066 0.8770 ~Z 0.0119 1.6645 0.0073 0.9551 )~3 0.0098 1.4740 -0.0008 -0.1116 )k 4 0.0060 0.9942 0.0008 0.1120

San Francisco ol 0.339 (-0.093, 0.791) 0.411 (-0.093, 0.936) P2 0.152 (-0.261, 0.525) 0.003 (-0.425, 0.425) )k I 0.0145 2.3712 0.0058 1.2558 h E 0.0163 2.5832 0.0064 1.4153 X 3 0.0106 2.0596 0.0038 0.8783 ~4 0.0102 1.9187 0.0046 1.0032

The expected rate of return due to thej th seasonal dummy is E[eXi] - 1. Since the biggest posterior mean of the ~,js is 0.021, and their variances are even smaller, the expected rate of return due to the j th seasonal dummy, E[eXJ] - 1, can be approximated by the posterior mean of )~j. Among the four cities, Atlanta experienced the biggest nominal constant term effect, with the rate of return due to the second quarter dummy reaching 2.1%, Chicago, on the other hand, has the lowest nominal constant term effect. San Francisco is the only city where the four constant term effects in the real index model are all positive.

A Bayesian analogue of the p-value associated with a hypothesis is the probability of the region where the posterior density is lower than the density under the null hypothesis. Denote this Bayesian analogue of the p-value by ~o. We can simply look at the value of ~o to test a hypothesis. A hypothesis is to be rejected if its ~o-value is lower than a chosen probability level. Table 7 reports the ~.o-values for the overall equality, pairwise equality, and jointly zero hypotheses of the seasonal dummy coefficients (01, 02, 03, 04). There do not exist strong differences among the returns due to seasonal dummies except for Chicago.

S E R I A L C O R R E L A T I O N A N D S E A S O N A L I T Y 159

Table 7. ~values for alternative hypotheses of the seasonal dummy coefficients.

Hypothesis Nominal Index Model Real Index Model

Atlanta )`1 = )`2 = )`3 = )`4 0 . 3 5 0 0 .831 )`l = )'2 0 . 7 9 7 0 .914 X1 = )`3 0 .388 0 .555 )`1 = )`4 0 .523 0 .828 )'2 = )`3 0 . 4 9 6 0 .711 )`2 = )`4 0 . 0 9 4 0 .463 )`3 = )`4 0 .828 0 .964 )`1 = )`2 = )`3 = )`4 = 0 0 .002 0 .348

Chicago )`1 = )`2 = )`3 = )`4 0 . 0 8 8 0 .123 )`1 = )`2 0 . 8 8 2 0 . 6 0 6 )`1 = )`3 0 . 2 2 4 0 .057 )`1 = )`4 0 .083 0 .042 )`2 = )`3 0 . 4 0 0 0 .227 )`2 = )`4 0 .012 0 .033 )`3 = )`4 0 .203 0 .355 )`1 = )`2 = h3 = )`4 = 0 0 . 0 0 4 0 .016

Dallas )`1 = )`2 = )`3 = )`4 0 .611 0 . 7 3 6 )`1 = )`2 0 .927 0 .957 )`1 = )`3 0 .763 0 .377 )`1 = )`4 0 .623 0 .652 )`2 = )`3 0 . 8 3 4 0 .522 )`2 = X4 0 . 1 8 6 0 .448 )`3 = )`4 0 .703 0 . 8 9 6 )`1 = )`2 = )`3 = h4 = 0 0 .018 0 .097

San Francisco )`1 = )`2 = )`3 = )`4 0 .547 0 .965 )`1 = )`2 0 .742 0 .917 )`1 = )`3 0 .416 0 .711 )`1 = )`4 0 .467 0 .846 )`2 = )`3 0 . 3 2 6 0 .655 )`2 = )`4 0 . 2 0 4 0 .741 )`3 = )`4 0 . 9 4 4 0 .887 )`1 = )`2 = )`3 = )`4 = 0 0 .005 0 .058

Note: ~o-value is the probability of the region where the posterior density is lower than the density under the null hypothesis.

For Chicago, the ~o-value for {P2 = / ) 4 } is less than 0.05 for both the nominal and real index models; that for {01 = P4} is less than 0.05 for the real index model, and less than 0.1 for the nominal index model. The ~o-values associated with the hypothesis of no seasonality (i.e., {01 = 02 = 03 = P4}) for Chicago are 0.088 and 0.123 for the nominal and real index models, respectively. Although the overall seasonality is not significant, the ~o- value for {01 = 02 = 03 = P4 = 0} is less than 0.02 in the nominal index model for each city, and is less than 0.1 in the real index model for Chicago, Dallas, and San Francisco.

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5. Conclusion

In this article, a Bayesian methodology is proposed to estimate the serial correlation and the seasonality of the price behavior of infrequently traded assets. The Bayesian method is found superior to the alternative two-step methods.

First, when assets are traded infrequently, as in the real estate market, the two-step, one- sample and two-step, two-sample OLS estimators of the AR(2) coefficients and ACFs will not only be biased, but also inconsistent. Second, while the two-step, two-sample IV estimator is consistent, it involves dividing the data into two samples, and thus introduces additional error. Finally, existing implementation of the two-step methods ignores the component of sampling error due to uncertainty about the true value of the index. In comparing the empirical results of alternative methods, it is found that the estimates are sensitive to different estimation techniques; this suggests that the choice of estimation method is important.

The empirical results based on the Bayesian model indicate that strong persistence of residential house price movement, measured either in nominal terms or in deflated real terms, exists in three of the four cities in the data set. The magnitude of serial correlation tends to be smaller (or more negative in the case of Atlanta) when measured in real sales prices than in nominal prices. The evidence supports the rejection of random-walk behavior in the real estate market.

Since the random-walk assumption is rejected, it would be desirable to estimate the real estate index with a more general model than has typically been used in earlier literature. An additional advantage of the Bayesian approach is that it can be used to estimate the market index as well (see Appendix 3).

The findings do not suggest strong seasonal differences except in Chicago. However, there is still a clear indication that the returns due to seasonal dummies are mostly strongly positive in the second quarter, with the first quarter following closeiy. Residential house prices tend to be relatively low in the third and fourth quarters. The hypothesis that the returns due to the seasonal dummies are jointly zero is rejected for each city in the nominal index model and for some cities in the real index model.

Acknowledgment

I would like to thank my advisors, Robert Shiller and Christopher Sims, for numerous suggestions. I would also like to thank Daniel Abrams, Abhijit Banerji, Paul Bergin, William Goetzmann, Roger Ibbotson, Peter Linneman, and Greg Sutton for helpful comments. The data for this study were provided by Robert Shiller.

Appendix 1

We demonstrate here that for given Pl and P2, (12) is a proper density of z. Denote the right-hand side of (12) by g(Pl, a2, z). To examine the upper limit, let

m(pl , P2, z) = z-(T-1)/2rI-(N-n-1)/2IxoMaXoI-'I/2 q(Pl, P2),


where

= (Y - XoO)'M,~(y - XoO )

q(Pl, /92) = Inl-l/2 (1 - /9, - /92) ( iT~- l iT ) 1/2"

Then limz_.o~ (g(Pa,/92, z)/m(pl, Oz, z)) = 1. Since faro(~91,/gz, z)dz is convergent for T > 3 and finite a, so is ~ag(/91,/91, z) dz.

For the lower limit, rewrite g(ol,/92, z) as

g(Pl , /92, Z) = Z 1/2 h-(N-n-1)/21M~[~-lMr + ZXoMaXol-1/Zq(/91, /02).

Since limz+oo g(Pl,/92, z) = 0 for finite a, it follows tha t f~g(p l , P2, z) dz is also finite.

Appendix 2

This appendix discusses the computation of XdM,~Xo, XdM~ Y and (Y - XoO)'M~(Y - X00 ). Assume that the data Y are arranged in order of houses. Then M~ is a block diagonal matrix with the ith block, denoted as mi, being a ki x k i matrix, of which the diagonal element is 1/ki and the off-diagonal elements (1 - ki)/ki, where ki is the frequency of sales for house unit i. Let X0i and Y/be the ith row of X 0 and Y, respectively; then

= XoimiXoi XoM~,X o i=1

X e M,~ Y = Xoi mi Yi i=1

(Y - Xo)'M,~(Y - Xo) = ~_a (Y - Xoi) 'mi(Y - Xoi). i=1

Appendix 3

Integrating cz, 4~ and Cr2u out of (11) gives

p(O, O1, 02, zIY)

0r z-(T-1)/21[~ I -1/2 h-(N+T-n-1)/2(1 _ O1 -- 192) ( i ~ - l i T ) 1/2

1 XdM,~Xo~ -I/2(N+ T-n-1)

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Therefore, given Ol,/)2, and z, 0 is a t-distribution with (N - n - 1) degrees of freedom, mean Co, and variance (h/(N - n - 3))((I/z) M~,~-IM4, + XdM~Xo) -1. Then the posterior mean of 0 can be computed by

t'll ( 1 - p 2 (OOCo g(Pl , /)2, Z) dzdpldp2 p(OIY) = J - 1 J p2-1 Jo

1 / ' 1 - 0 2 t ' ~176 f-, L 2 - 1 L g(Pl , I)2, Z )dzdp ldp2

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Serial correlation and seasonality in the real estate market