1. Journal of Real Estate Finance and Economics, 16: 1, 75±90 (1998)
# 1998 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
Developing Con®dence Intervals for Of®ce
Market Forecasts
RICHARD K. GREEN
University of Wisconsin±Madison
STEPHEN MALPEZZI
University of Wisconsin±Madison
WALTER BARNES
University of Wisconsin±Madison
Abstract
This study focuses on the precision of models that forecast of®ce construction and absorption. The article is novel
because for the ®rst time it applies Feldstein's (1971) technique for developing forecast standard errors in the
presence of stochastic exogenous variables. The purpose of the article is not to ®nd behavioral relationships but
rather to evaluate forecasts. We ®nd that in the case of many of®ce markets, standard errors of long-term forecasts
for absorption and completions are quite large, and therefore the forecasts themselves should not be used as a
reliable basis for underwriting.
Key Words: Of®ce Market, Forecast, con®dence interval
1. Introduction
Real estate ®rms and ®nancial institutions often use time-series models for forecast of®ce
market rents, construction, and absorption. These models provide some of the information
required to value, underwrite, and in other respects analyze of®ce buildings.
Firms that use these models undoubtedly have a better information set for making real
estate decisions than ®rms that do not. Yet a question remains as to how much these
models do, in fact, improve the information set. Put another way, little past work has been
done on using analytical techniques to measure the precision of of®ce market forecasting
models. One notable exception is the work of Goetzmann and Wachter (1992), who have
used bootstrap methods to place con®dence bands about association frequencies to
determine the relationships between movements of rents and vacancies across different
cities.
Our study examines the precision of models that forecast of®ce construction and
absorption. We believe such a measure can be useful to lenders and developers considering
the possibility of ®nancing and constructing speculative buildings or preleased buildings
containing short-term leases. If we can demonstrate that forecasting models display a great
dealÐor at least a relatively constant amountÐof precision, they can be relied on with
2. some degree of con®dence. If, on the other hand, they become increasingly imprecise at a
rapid rate as the forecast period gets longer, they will not be particularly reliable bases for
underwriting. This article is novel because for the ®rst time it applies Feldstein's (1971)
technique for developing forecast standard errors in the presence of stochastic exogenous
variables. The purpose of this article is moreover not to ®nd behavioral relationships but
rather to evaluate forecasts. We therefore focus on a reduced-form model. We ®nd that at
least in the case of the Atlanta of®ce market, standard errors of long-term forecasts for
absorption and completions are quite large.
2. The Model
A number of scholars (e.g., Clapp, Pollakowski, and Lynford, 1992; Shilling, Sirmans, and
Congel, 1987; Wheaton, 1987) have developed behavioral models of the of®ce market.
These models have been valuable for forecasting output as well as for estimating such
derivative outcomes as the natural vacancy rate and, more generally, the elasticity of the
demand for of®ce space with respect to of®ce employment and rents.
Our purpose here, however, is not to investigate behavioral relationships but rather to
develop forecasts and to measure their accuracy and precision. To that end, we focus on
reduced-form models.
Even behavioral models such as Wheaton and Torto's must be converted to reduced
forms when they are used for forecasting purposes. Therefore, we directly estimate
reduced-form models for of®ce absorption, for of®ce construction completions, and for
of®ce employment. We use past behavioral models to help guide us in our selection of
explanatory variables.
Our forecasting strategy is to estimate equations suggested by the Granger causality
literature. An explanatory variable is said to Granger-cause a dependent variable if and
only if lagged values of the explanatory variables combined with lagged values of the
dependent variables forecast the dependent variable better than the lagged values of
the dependent variables alone. Because our only purpose here is to present a method for
evaluating a forecast, we believe that our forecasting approach has some intuitive appeal:
unless a variable helps us improve our forecast, we do not use it. This allows us to specify
parsimonious models, which, as we shall see, should help reduce our forecast variance.
The focus of our analysis is Atlanta, a large market that, as we shall see, is fairly typical
and works well for forecasting purposes. We also brie¯y consider other cities to investigate
the robustness of our conclusions.
Clapp (1993) provides an excellent review of behavioral models of the supply of and
demand for of®ce space. He highlights a number of stylized facts that we incorporate in
searching for our ``best'' reduced-form equationsÐthat demand and therefore absorption
is largely a function of of®ce employment; that supply responds with a long lag to shifts in
demand, vacancy, and rent; and that vacancy rates ultimately tend to return to some natural
rate. We therefore investigate whether such variables as of®ce employment, vacancy, and
the size of the of®ce stock in a market Granger-cause of®ce construction and of®ce
absorption.
76 GREEN, MALPEZZI, AND BARNES
3. We use a straightforward model to estimate the following set of equations for the
Atlanta metropolitan area:
ABSt ˆ a1 ‡ B11ABStÀ1 ‡ Á Á Á ‡ B1nABStÀn ‡ g11ZtÀn ‡ d1T ‡ e1 …1†
CMPt ˆ a2 ‡ b21CMPtÀ1 ‡ Á Á Á ‡ b2nCMPtÀn ‡ g21ZtÀ1 ‡ Á Á Á ‡ g2nZtÀn
‡ d2T ‡ e2 …2†
and
OEMPt ˆ a3 ‡ b31OEMPtÀ1 ‡ Á Á Á ‡ b3nOEMPtÀng31ZtÀ1
‡ Á Á Á ‡ g3nZtÀn ‡ d3T ‡ e3Y …3†
where ABS is absorption, CMP is completions, OEMP is of®ce employment, Z represents
other explanatory variables, T is time (in this case, six-month periods), and the e's are true
error terms. The vector Z represents any other variables found to Granger-cause any of
the endogenous variables, as is discussed below.
To perform forecasts of the size of the of®ce stock and vacancies, we also employ
these identities:
Stockt StocktÀ1 ‡ CMPtÀ1 À RemovalstÀ1 …4†
and
Vacancyt VacancytÀ1 À
ABSt À CMPt
StocktÀ1
X …5†
Because we have no independent data on removals, we approximate these as 2.5% of
the existing stock, following Hulten and Wykoff (1981). Because equations (1) to (3) are
reduced-form forecasting equations (albeit ones that should re¯ect the interaction of
supply and demand characteristics), we use a simple rule for determining which
explanatory variables we keep in our ®nal forecast equations. If we cannot reject the
hypothesis that an explanatory variable Granger-causes a dependent variable at the 90%
level of con®dence, we retain the explanatory variables.
We should underline a few other characteristics of equations (1) to (3). First, note that
we do not test whether rent Granger-causes completions or absorption. As we discuss
below, we omit rent from our equations simply because we have a very limited number of
observations on rent. Second, we have decided to employ an ARIMA model for
forecasting of®ce employment. We use our own ARIMA model, rather than a third-party
model of of®ce employment, so that we may get an estimate of the overall level of
variance associated with the forecasts we perform using equations (1) and (2). Under many
circumstances, ARIMA models seem to forecast as well as any others.1
We have a prior belief that shocks to local economies will affect employment,
completions, and absorptions contemporaneously. Under such conditions the method of
DEVELOPING CONFIDENCE INTERVALS FOR OFFICE MARKET FORECASTS 77
4. seemingly unrelated regressions (SUR) will yield more ef®cient estimates than OLS. The
SUR method uses the following equation for estimating coef®cients:
b ˆ …XH
SÀ1
IX†À1
XH
SÀ1
IXH
yY …6†
where y consists of the vectors of dependent variables stacked on each other from (1) to
(3), and X is
X1 0 0
0 X2 0
0 0 X3Y
…7†
where XiY i ˆ 1Y 2Y 3 indicates the exogenous variables in equation (i). The matrix S is a
3-by-3 matrix of the variance of and covariance between the residuals of the three
equations. The denotes the Kronecker product operator. The variance for an equation
(i) is estimated with
si ˆ
eH
iei
…n À ki†
Y …8†
where ei is the vector of residuals from the estimated equation (i), and ki is the number of
explanatory variables in equation (i). To estimate the covariance of the residuals between
the equations (i) and ( j), we calculate
sij ˆ
eH
iej
…n À ki†…n À kj†
q X …9†
As already noted, the SUR technique, in a properly speci®ed model, gives us narrower
con®dence intervals than OLS. Also, because the SUR model allows us to estimate the
covariance of the disturbances of our equations, it allows us to estimate the variance of
the vacancy forecasts, for reasons that will become apparent below.
We make no claim that the forecasts we produce using the procedure just described are
any better than anyone else's, although we have no reason to think they will be any worse.
They permit us to determine whether there is a pattern to the variance of forecasts that is
worthy of further exploration.
Once we estimate equations (1) to (3), we may use them (and, if necessary, identities (4)
and (5)) to forecast of®ce space construction, absorption, and of®ce employment. The next
step is to determine the standard error of the forecasts.
For ®tted values, or forecasts, where all explanatory variables are ®xed (that is, non-
stochastic), the variance of the forecast is simply
s2
”y ˆ s2
‡ s2
xj…XH
…S I†À1
X†À1
xH
jY …10†
78 GREEN, MALPEZZI, AND BARNES
5. where xj is the vector of explanatory variables used to forecast y. The other terms are
de®ned as before. This familiar expression reminds us that one source of error is due to
our forecast assumption that each period's error is zero when of course it is (at best) on
average zero; hence the ®rst s2
term. The longer expression to the right of the ®rst s2
reminds us that another source of forecast error is our use of coef®cient estimates in place
of the true parameter values. These two sources can be denoted forecast error and
regression error, respectively.
In our context, however, there is a third source of error, which is sometimes denoted
conditioning error. This error arises in cases where the value of X on which the forecast is
conditioned is stochastic. Clearly, this is the case in our time-series forecasting models
(and virtually all others), since forecasts made beyond period t ‡ l are, of course, functions
of other forecasts. All such forecasts are consequently derived from the sums of the
products of two random variables. This fact has important consequences both for the
estimation of standard errors and the distribution of the errors of the forecasts.2
Feldstein (1971) develops formulae for the standard error of a forecast, and in particular
discusses the problem of de®ning forecast intervals in the presence of conditioning error
(that is, when exogenous variables are forecast and hence stochastic). Let us de®ne the
entire vector of the reduced-form least-square coef®cients as p, and the variance-
covariance matrix of the coef®cients as G3
Ðthat is,
G ˆ s2
…X
H
…S I†À1
X†À1
X …11†
We will also de®ne a variance-covariance matrix of the forecast-period explanatory
variables as OÐthat is,
O ˆ s2
xF…X
H
…S I†À1
X†À1
xFY …12†
where xF denotes the forecast period explanatory variables. Note that the ®rst element of
the diagonal of O, which is the variance of the constant, will always be zero. The second
element of the diagonal will be equal to the variance of the ®rst explanatory variable
other than the constant, and so on.
We assume that the estimates of the regression coef®cients and the forecast-period
exogenous variables are uncorrelated: E‰…”xF À xF†…”p À p†Š ˆ 0. The hat above a variable
denotes that it is an estimate. The fact that we use lagged dependent variables on the right
sides of the estimation equations perhaps casts a shadow on the validity of this assumption.
As a practical matter, however, we have no good way to estimate the covariance of the
coef®cients and the forecast period exogenous variables.
Feldstein shows that, given the assumption above, we may de®ne the forecast variance
for period r as
s2
”yr
ˆ s2
‡ x
H
FGxF ‡ p
H
Op ‡ trace…GO†X …13†
The ®rst two terms of the RHS of equation (13) are equivalent to (10). The last two
DEVELOPING CONFIDENCE INTERVALS FOR OFFICE MARKET FORECASTS 79
6. terms are due to conditioning error, and the measured variance of a forecast correctly
incorporating conditioning error will exceed measures that neglect this source.
In estimating our regression coef®cients, we also estimate G. But how to estimate O is
less obvious. We conjecture that one reason Feldstein's procedure has been somewhat
neglected in applications is that his paper does not explain how to estimate O. We estimate
O recursively. For our forecast in period t ‡ 1, the matrix we call Ot‡1 is simply a matrix of
zeros because we know the values of all variables from tY t À 1Y F F F Y t À n. Therefore, we
®nd the variance of the forecast in the usual way, using the ®rst term in (13).
In forecast period t ‡ 2, however, we base our forecast on some variables from t ‡ 1Ð
that is, forecast variables. We may use our estimate of the forecast variance from period
t ‡ 1 to estimate values in the appropriate places for what we may call Ot‡2; this allows us
to estimate the variance of the period t ‡ 2 forecast. We then use the variance calculated
for t ‡ 2 to get Ot‡3 necessary for estimation of the forecast variance in t ‡ 3 and so on.
Feldstein gives a numerical example showing how the second and third terms on the right
side of (11) can be very important. We use this procedure to calculate variances for our
forecasts of completions, absorptions, and of®ce employment.
Finding the variance of the forecast of®ce stock and vacancy rate is far more dif®cult
because it involves calculating the covariances of forecasts across periods. Consequently,
for now we will calculate ®rst-order approximations of the forecast variance of the stock
and vacancy rate. By ignoring covariance terms, for the variance of the stock forecast
we have
s2
stockYt‡j ˆ
ˆt‡j
iˆt‡1
s2
cmpà …14†
To ®nd a ®rst-order approximation of the variance of the vacancy forecast, we have4
s2
vacYt‡j ˆ
ˆt‡j
iˆt‡1
…C ”MPi À A ”BSi†
2
St”ock2
i
Ã
s2
cmpYi ‡ s2
absYi À 2s12
…C ”MPi À A ”BSi†2
‡
s2
stockYi
St”ock2
i
2 3
X …15†
The notation has the obvious meaning, except for s12, which is the covariance between
the regression equations for absorption and completions.
The fact that our forecasts are the results of the products of two random variables also
has an important distributional implication: we cannot assume that our forecast errors are
normally distributed and therefore cannot calculate con®dence intervals in the usual
manner. Indeed, as Feldstein notes, we cannot even assume that the distribution of the
errors is unimodal or symmetric. Conservative con®dence intervals can be developed
using the Tchebychev inequality, which says that for a forecast value y
P‰j”yF À yFj ! ks”yF
Š 1ak2
X …16†
Feldstein points out that we can read equation (14) in the following way: the probability
80 GREEN, MALPEZZI, AND BARNES
7. that the observed value of y in the forecast period will fall outside the interval Æks does
not exceed 1ak2
. This is similar in spirit to a Bayesian con®dence interval. For comparison,
we also construct the con®dence interval in the ``usual'' wayÐthat is, Æ2 standard errors.
The unknown true con®dence interval will be bounded by these two intervals.
3. The Data
Our data come from the CB Commercial/Torto Wheaton Of®ce Market Data Service. The
data are available for ®fty-one markets in the United States and contain semiannual data on
of®ce employment growth, population growth, vacancy, absorption, completions, stock,
and a rental index for each market. All data are available semiannually from 1967 through
1992, except for the rental index, for which data are only available for the years 1980
through 1992. Because of the limited number of observations on of®ce market rents, we
omit a rent-forecasting equation from our study. We use all twenty-six years available to us
for estimating our forecast models. This has implications that we will discuss below.
The methods for collecting and verifying the data are described in CB Commercial/
Torto Wheaton Research (1992).
4. Pretests of the Data
Our statistical analysis maintains certain classical assumptions that are often violated with
time-series data. In this section we brie¯y discuss some of the tests undertaken prior to
analysis.5
We began by plotting autocorrelation functions and computing augmented Dickey-
Fuller tests for unit roots for each data series.6
Completions and employment were clearly
not stationary; only for absorption could we reject the hypothesis of a unit root (consistent
with an assumption of stationarity).
However, two or more series that are individually nonstationary may be cointegrated,
in which case error terms from a regression equationÐa linear combination of the
variablesÐwill be stationary.7
It is the stationarity of the error term that is required for
validity of our additional tests.
The test statistic for the cointegrating vector of our three variables (plus a trend term) is
4.14. MacKinnon's critical values are 4.32 at 5% and 3.99 at 10%. Our interpretation of
this statistic is that we can reasonably reject the unit root hypothesis for the error term, but
we do note there is somewhere between a one in ten and one in twenty chance that we have
rejected the null when it is in fact true (that the error term is nonstationary).
It is worth pointing out that both absorption and completions are ¯ow measures of of®ce
market activity. As a reviewer correctly pointed out, stock models would also be of
interest. One disadvantage of such models, however, is that error terms from such models
will certainly be far from stationary and the interpretation of estimated variances
problematic.
DEVELOPING CONFIDENCE INTERVALS FOR OFFICE MARKET FORECASTS 81
8. 5. Estimation Results
After performing a series of Granger tests, we found that the only variable within our data
set that Granger caused either absorption or completions is of®ce employment: we also
found that employment only weakly caused absorption. We also used F-tests to determine
the number of lags we used in our ®nal forecasting models. The estimation results we used
in our forecasting models are attached as Appendix A.
Our forecast estimates, their standard errors, and their component parts are presented in
tables 1 through 3. Note that the standard error of the regression in our context provides by
far the largest portion of the forecast standard error, suggesting that the two terms from
equation (13) that were so important in Feldstein's (1971) example are not so important to
us here. In particular we note the ®nal column, which is the variance from conditioning
error, is quite small. We also note that while the variance of the forecast increases over
time, the increase is slow. First, one major component of the forecast variance, from the
regression estimates, is ®xed by construction. Some readers may be initially puzzled by
the fact that the overall forecast variance increases so slowly. Every textbook emphasizes
(correctly) that forecast variances increase as one moves out of sample. But ``out of
sample'' in this model means forecast xF very different from in sample xj and (except for
the time trend) our forecasts xF are generally quite close to observed values of x. Finally,
the remaining variance from conditioning error does increase over time but is dominated
by the other two sources.
Table 1. Atlanta: Completions forecast, and variance of forecast, by component.
Period
Completions
Forecast
Forecast
Variance
Regression
Variance
Conditioning
Variance
Total
Variance
of Forecast
Standard
Deviation
of Forecast Year
1 1,854 272.442 28 0 272,470 522 1993.0
2 2,133 272,442 30 290,478 562,950 750 1993.5
3 2,597 272,442 35 654,513 926,991 963 1994.0
4 3,011 272,442 48 1,087,244 1,359,734 1,166 1994.5
5 3,328 272,442 64 1,592,871 1,865,377 1,366 1995.0
6 3,478 272,442 84 2,175,478 2,448,003 1,565 1995.5
7 3,481 272,442 99 2,840,270 3,112,811 1,764 1996.0
8 3,378 272,442 109 3,593,451 3,866,002 1,966 1996.5
9 3,233 272,442 110 4,442,357 4,714,909 2,171 1997.0
10 3,101 272,442 106 5,395,353 5,667,902 2,381 1997.5
11 3,025 272,442 99 6,462,018 6,734,559 2,595 1998.0
12 3,020 272,442 92 7,653,246 7,925,780 2,815 1998.5
13 3,078 272,442 88 8,981,350 9,253,880 3,042 1999.0
14 3,179 272,442 88 10,460,050 10,732,579 3,276 1999.5
15 3,295 272,442 90 12,104,902 12,377,435 3,518 2000.0
16 3,401 272,442 96 13,933,049 14,205,587 3,769 2000.5
17 3,482 272,442 102 15,963,701 16,236,245 4,029 2001.0
18 3,531 272,442 108 18,218,385 18,490,935 4,300 2001.5
19 3,553 272,442 113 20,720,886 20,993,441 4,582 2002.0
20 3,559 272,442 116 23,497,675 23,770,233 4,875 2002.5
82 GREEN, MALPEZZI, AND BARNES
10. We may rely on the Tchebychev inequality to know that there is less than a 6% chance
that the true value of a variable we are attempting to forecast is more than four standard
errors away from our forecast value. Thus we graph the forecasts surrounded by both two
standard deviations (the 95% con®dence interval under the assumption of normality) and
four standard deviations.
First look at ®gure 1, which graphs the completions forecast (middle line), as well as
lower-bound and upper-bound con®dence intervals of the forecast for Atlanta. The lower-
bound con®dence interval is the 95% con®dence interval for the forecast assuming that the
errors are normally distributed. This is the narrowest possible con®dence band. The upper-
bound con®dence interval is the 93.75% interval arising from the Tchebychev inequality.
This is the widest-possible con®dence band.
The forecast itself may be characterized as arising from a pattern of damped oscillation,
re¯ecting the fact that we have a second order autoregressive process (see Box and
Jenkins, 1976, p. 58). Note that the number of completions forecast rises quickly over the
®rst few forecast years. Atlanta over the past few years has been a fairly typical of®ce
market by national standards: overbuilding in the middle to late 1980s has caused a near-
depression in of®ce construction. Any time-series autoregressive forecast such as ours,
which uses a fairly long data series (in our case, ®fty-one observations), will therefore
project construction to return fairly rapidly to levels commonly seen over the period from
which the coef®cients are estimated.
Figure 1. Atlanta completions forecast.
84 GREEN, MALPEZZI, AND BARNES
11. To a certain extent, however, our con®dence bands re¯ect our skepticism about whether
of®ce construction will come back in Atlanta as rapidly as the forecast model would
suggest. If we look at the upper-bound con®dence interval, which is the ``Bayesian''
con®dence interval at the 93.75% level of con®dence,8
we can see that it remains entirely
possible that of®ce construction in Atlanta will remain subdued for several years to come.
Still, even employing the most pessimistic con®dence band as the basis for our forecast,
the model predicts that Atlanta will over the longer run see of®ce construction return to
and remain at levels more customary than those of the last few years.
Absorptions tell much the same story (see ®gure 2). Once again, the model exhibits
damped oscillation and is clearly strongly in¯uenced by history: it suggests absorptions
will rapidly return to something like the mean levels of the past twenty years. A rapid
increase in absorption seems more plausible to us than a rapid increase in construction
activity; nevertheless, tests of coef®cient stability are necessary to determine whether we
should in fact use all years from our data set from our forecast model. Moreover, the
con®dence bands once again suggest that it is well within the realm of possibility that
absorption will not quickly rebound to historical levels.
Despite our lack of comfort at this stage with the completions forecast, the vacancy
forecast, which predicts slowly declining vacancy rates over the next ®ve years, seems
eminently reasonable to us (®gure 3). Our con®dence however, can hardly be bolstered by
our estimates of ®rst-order approximations of standard errors, which show the con®dence
Figure 2. Atlanta absorptions forecast.
DEVELOPING CONFIDENCE INTERVALS FOR OFFICE MARKET FORECASTS 85
12. bands around the vacancy forecasts to become quite wide relative to the variable we are
forecasting. The out-year forecasts suggesting negative vacancy are also less than
reassuring. Whether this outcome would be exacerbated if we were to include higher-order
terms in our estimate of variance remains to be seen. The relatively small impact of the
second and third terms from the right side of (11) suggest that leaving the higher-order
terms out should be fairly innocuous.
We should emphasize that even though the standard error of the vacancy forecast
relative to the mean is fairly small in absolute terms, the difference between our forecast
vacancy rate and most pessimistic con®dence bound is even in early years as high as
100%. Needless to say, for those developing or underwriting loans for of®ce buildings, this
is not a trivial difference.
Finally, our employment forecast (®gure 4), simple though it is, seems to perform quite
well.9
Even our outer-limit con®dence bands are relatively close to our forecast values.
6. Other Cities
We ran the model we speci®ed for Atlanta for several other large cities, including Boston,
Chicago, Dallas, Denver, Houston, Los Angeles, New York, and Washington, DC. To get a
Figure 3. Atlanta of®ce vacancy forecast.
86 GREEN, MALPEZZI, AND BARNES
13. sense of how the con®dence intervals of these forecasts compared with Atlanta's, we
report standard errors for one-, ®ve-, and ten-year vacancy-rate forecasts in table 4.
A few results strike us as particularly interesting. For one-year forecasts, all cities have a
standard error of 1%. The con®dence intervals of longer-term forecasts, however, varies
substantially across cities. The Washington, DC forecast's standard error remains
Figure 4. Atlanta of®ce employment forecast.
Table 4. Standard errors for three vacancy forecasts in ten selected cities.
One-Year
Standard Error
Five-Year
Standard Error
Ten-Year
Standard Error
Atlanta 0.01 0.03 0.07
Boston 0.01 0.03 0.05
Chicago 0.01 0.03 0.08
Dallas 0.01 0.04 0.09
Denver 0.01 0.04 0.10
Houston 0.01 0.05 0.13
Los Angeles 0.01 0.03 0.10
New York 0.01 0.03 0.06
San Francisco 0.01 0.03 0.07
Washington, DC 0.01 0.03 0.04
DEVELOPING CONFIDENCE INTERVALS FOR OFFICE MARKET FORECASTS 87
14. relatively small at 4% (although we should also note that this produces a con®dence band
somewhere in between 16 and 32%). Houston, on the other hand, has a forecast standard
error that rises to 5% after only ®ve years. This outcome seems quite sensible in light of the
fact that Washington's of®ce employment base (government workers and lobbyists) is
quite stable, while Houston's (resource-based industry) is not.
7. Conclusion
In this article we have attempted to take the ®rst steps necessary for developing con®dence
intervals for of®ce market forecasts. Unlike past work, we use Feldstein's (1971)
technique for determining forecast standard errors when the exogenous variables are
themselves stochastic, and use Tchebychev's inequality to develop con®dence intervals.
Future revisions and extensions include using in-sample techniques to develop empirical
con®dence intervals, tests of coef®cient stability, and re®nement of the estimates of the
variances of the of®ce stock and the vacancy rates. We believe the development of
con®dence intervals for forecasts is important because real estate ®rms and ®nancial
institutions must have a true sense of how markets might perform under the most
pessimistic of circumstances.
Appendix A. Regression Results for Atlanta Forecast
COMP ABS OEMP
Constant À 142 À 339 7.41
(493) (649) (3.36)
COMPÀ1 0.96
(0.12)
COMPÀ2 0.38
(0.12)
ABSÀ1 0.19
(0.13)
ABSÀ2 0.08
(0.12)
ABSÀ3 À 0.33
(0.12)
OEMPÀ1 43.6 108 0.93
(21.4) (29) (0.05)
OEMPÀ2 À 38.9 À 75.3
(20.9) (37.6)
OEMPÀ3 À 24.1
(19.1)
T À 11.2 À 37.8 0.66
(53.9) (70.8) (0.37)
R2
0.751 0.471 0.995
88 GREEN, MALPEZZI, AND BARNES
15. Appendix B. Variance±covariance Matrix of Completions, Absorptions, and Of®ce
Employment Equations
COMP ABS OEMP
COMP 274531 168067 418
ABS 452965 À 20.2
OEMP 13.0
Acknowledgments
We wish to thank Ray Torto for granting us permission to use the CB Commercial/Torto
Wheaton database for this project. We are also grateful to John Clapp and two anonymous
referees for useful comments. The usual caveat certainly applies.
Notes
1. For example, Cooper (1972) showed that autoregressive equations of macroeconomic variables provide better
one-step ahead forecasts than well-known simultaneous equation structural forecasting models.
2. Of course, one other source of error comes from the inevitable misspeci®cation of any econometric model. But
this cannot be measured.
3. Our notation differs somewhat from Feldstein's. Our i does not appear in his article because he does not
consider the SUR technique. Our G is his O, and our O is his i.
4. For the derivation of this calculation, see Mood, Graybill, and Boes (1974).
5. Many discussions of these tests are available. See, for example, Granger and Newbold (1986) or Pindyck and
Rubinfeld (1991).
6. Critical values for the test are taken from MacKinnon (1991) from tables provided in MicroTSP.
7. More precisely, if the true error terms are stationary, and if we have the correct model, the estimated errors (the
residuals) should exhibit stationary behavior.
8. That is, Æ4 standard deviations away from the forecast values.
9. The lines in the of®ce employment forecast graph are not labeled because they are so closely spaced. They are
analogous to the lines in the three other graphs.
References
Box, George E. P., and Gwilym M. Jenkins. (1976). Time-Series Analysis: Forecasting and Control. Holden-Day.
CB Commercial/Torto Wheaton. (1992). The Of®ce Outlook Report.
Clapp, John M. (1993). ``Dynamics of Of®ce Markets: Empirical Findings and Research Issues,'' Washington,
D.C. AREUEA Monograph Series No. 1.
Clapp, John M., Henry O. Pollakowski, and Lloyd Lynford. (1992). ``Intrametropolitan Location and Of®ce
Market Dynamics,'' Journal of the American Real Estate and Urban Economics Association 20(2), 229±259.
Cooper, R. L. (1972). ``The Predictive Performance of Quarterly Econometric Models of the United States.''
In B. G. Hickman (ed.) Econometric Models of Cyclical Behavior. New York: Columbia University
Press.
DEVELOPING CONFIDENCE INTERVALS FOR OFFICE MARKET FORECASTS 89
16. Feldstein, Martin. (1971). ``The Error of Forecast in Econometric Models When the Forecast Period Exogenous
Variables Are Stochastic,'' Econometrica 39(1), 55±60.
Goetzmann, William, and Susan Wachter. (1992). ``Bootstrapping Con®dence Bands About Association
Frequencies: Applying Clustering Methods to Commercial Rents.'' Working paper.
Goldberger, Arthur. (1991). ``A Course in Econometrics.'' Cambridge: Harvard University.
Granger, C. W. J., and Paul Newbold. (1986). Forecasting Economic Time Series (2nd ed.). New York: Academic
Press.
Hulten, Charles R., and Frank C. Wykoff. (1981). ``The Measurement of Economic Depreciation.'' In Charles R.
Hulten (ed.), Depreciation, In¯ation and the Taxation of Income from Capital. Urban Institute.
MacKinnon, James G. (1991). ``Critical Values for Cointegration Tests.'' In R. F. Engle and C. W. J. Granger
(eds) Long-Run Economic Relationships: Readings in Cointegration. New York, Oxford University Press,
1992.
Mood, A. M., F. A. Graybill, and D. C. Boes. (1974). Introduction to the Theory of Statistics. New York:
McGraw-Hill.
Pindyck, Robert S., and Daniel L. Rubinfeld. (1991). Econometric Models and Economic Forecasts (3rd ed.).
New York: McGraw-Hill.
Shilling, James D., C. F. Sirmans, and John B. Corgel. (1987). ``Price Adjustment Process for Rental Of®ce
Space,'' Journal of Urban Economics 22(1), 90±100.
Wheaton, William C. (1987). ``The Cyclic Behavior of the National Of®ce Market,'' Journal of the American
Real Estate and Urban Economics Association 15(4), 281±299.
90 GREEN, MALPEZZI, AND BARNES
