Improving the Forecast Accuracy of Provisional Data: An Application of the Kalman Filter to Retail Sales Estimates

International Finance Discussion Papers Number 318

December 1987

IMPROVING THE FORECAST ACCURACY OF PROVISIONAL DATA:

AN APPLICATION OF THE KALMAN FILTER TO RETAIL SALES ESTIMATES by

B. Dianne Pauls

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an

acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.

ABSTRACT

If forecasts of economic activity are to rely on preliminary data, the predictable component of the data revisions should be taken into account. This paper applies the Kalman filter to improve the forecast accuracy of published preliminary estimates of retail sales. Successive estimates of retail sales are modeled jointly as a vector autoregressive process, incorporating panel rotation and calendar effects. Estimates of retail sales based on this model are then combined with the raw Census estimates via the Kalman filter. This technique, which may be applied to other bodies of data, yields a significant improvement in the efficiency of the

raw Census data, reducing the mean-squared error by about 1/3.

Improving the Forecast Accuracy of Provisional Data:

An Application of the Kalman Filter to Retail Sales Estimates

by

B. Dianne Pauls!

1. Introduction

Numerous studies have documented the substantial nature of revisions to macroeconomic data.” Because timely forecasts of economic activity often rely on such preliminary data nonetheless, assessing whether the data revisions are forecastable is crucial. A recent study by Mankiw and Shapiro (1986) suggests that, although revisions to GNP are quite large, they can not be forecast based on known information. Thus, the preliminary data represents the best forecast -- in a mean- squared error sense -- of the final estimate given all the available information, and their literal use macroeconomic projections produces efficient forecasts. This result contrasts with earlier work by Howrey (1978) and Conred and Corrado (1978) which indicates that revisions to disposable income and retail sales are forecastable based on known information.

These studies further demonstrate substantial scope for improving

1. The author is a member of the Division of International Finance, Board. of Governors of the Federal Reserve System. This work was begun while the author was in the Division of Research and Statistics. The views expressed herein are the author’s own and do not necessarily reflect the views of the Board of Governors or other members of its staff. I have benefited from conversations with Susan Burch, William Cleveland, Neil Ericsson, and several analysts at the Bureau of the Census and from comments by David Pierce on an earlier draft. I am also grateful to Stephen Gordon and William Nelson for research assistance.

2. See for example, Zellner (1958), Cole (1969), and Mankiw and Shapiro (1986).

forecasting accuracy by using signal extraction methods to exploit: this correlation.

This paper follows the work of Conrad and Corrado in using the Kalman filter to derive a better estimate of underlying retail sales activity. Retail sales are a prime candidate for this analysis because the revisions often are sizable relative to the movement in the underlying series. Over the period of this study -- January 1982 to May 1986 -- the mean absolute revision in the month-to-month change in total retail sales (excluding autos, gasoline, and nonconsumption items) is 0.46 percent, while the mean absolute month-to-month change is 0.73 percent. Furthermore, an examination of the Census Bureau's sampling procedures suggests a likely correlation in the data revisions. In particular, monthly estimates of retail sales are based on a rotating panels of stores. Thus, if these panels are distinct, one would expect, at a minimum, a correlation between estimates of sales activity based on a common panel of stores.

The Kalman filter provides an algorithm for signal extraction; the efficiency of the preliminary data is improved by combining it with an alternative information source. The alternative information is the “prediction of the final estimates based on a time series model that explicitly incorporates. the structure of the data revisions. The resulting estimates are more efficient in the sense of having a smaller mean-squared error than either the preliminary observation or the modelbased prediction.

This paper differs from earlier work by Conrad and Corrado primarily in its approach to the time series modeling of retail sales. A

more explicit model of the panel rotation bias in retail sales is.

presented. And, rather than estimate univariate models using seasonally differenced data, successive estimates of retail sales are modeled jointly as a vector autoregressive process. The parameters of this process are estimated simultaneously with calendar effects, such as the composition of the month and the occurrence of Easter. In addition, because the Census Bureau substantially revised its sampling procedures in October 1977, the time series model presented in Conrad and Corrado probebly is no longer applicable.

The results yield a significant improvement in the efficiency of the raw Census data, reducing the mean-squared error by about 1/3. Given that retail sales are the primary source data for personal consumption expenditures -- the largest component of GNP -- there are two possibilities for reconciling these findings with those of Mankiw and Shapiro. Either the Bureau of Economic Analysis takes account of the systematic nature of revisions to retail sales data in constructing quarterly personal consumption figures, or revisions to other components of GNP behave in such way as to offset this correlation.

The paper begins with a general discussion of the application of the Kalman filter to preliminary data -- contained in Section 2. Before modeling the data revision process, however, an understanding of the Census Bureau’s construction of successive estimates of retail sales is required. Section 3 outlines these procedures. A joint time series model of the final estimates and the data revisions is presented in Section 4. Once the systematic components of the revisions have been incorporated into the model, the Kalman filter is applied in Sections 5 and 6 to reduce the non-systematic components of the revision process.

Section 7 contains concluding remarks.

2. The Application of the Kalman filter preliminary data

The Kalman filter provides an algorithm for using the dynamic behavior of a time series to reduce the measurement error in the sample observations of that series. It entails a set of equations that utilizes the information in the latest, possibly error-ridden, observation to update the prediction of the true data based on its own past values and, perhaps, other exogenous variables. The application of the Kalmen filter to preliminary data is just a special case of the measurement error problen.

The first step in applying the Kalman filter is to specify the evolution of the final series -- the data without measurement error.

This dynamic behavior is described in the so-called state or trar.sition

equation. Denoting the final data as Xe» the state equation is written

as a first-order Markov process:

x7 AX, 4 + Cz, + Re, (1) where x, is a vector representing the state at time t; and A, C, and R

are matrices of known coefficients, z, is a vector of exogenous

t ‘variables, and e, is a vector of random disturbances with mean 0 and covariance matrix =. Although the state equation is specified as a first-order Markov process, the model can easily be extended to encompass higher-order autoregressive processes for the data by suitably augmenting the state vector. .

Next, the observation or measurement equation relating the

preliminary data to the final data is specified. Letting y, denote the

preliminary data, this equation is:

y= Hx, tv, (2)

where yY, is a vector of preliminary observations, of possibly smaller dimension than Xe H is a matrix of known coefficients, and v, isa vector of observation equation errors with mean 0 and covariance matrix V.

The one-period prediction problem is to forecast the final data for date t, Xe conditional on the information available at time t, which

includes the preliminary data, Yer the exogenous variables, and

Zh previous estimates of the state, Xe: Under the assumption that e, and Vv, are mutually uncorrelated, serially independent multivariate normal processes and the loss function is symmetric, the optimal predictor, Xelt is given as (Harvey (198la, 1981b)):

Fele ~ Sefe-a * Ke Ge - BReye-v (3)

re: = + i icti where Se] t-1 AXe a t-1 Cz. is the prediction of x based on the state equation. Denoting the covariance matrix of the state estimate errors,

Xx, - Xele-1? as Pelt-1? the Kalman gain, K,, can be expressed as:

-1

H'(V + HPL, 4H")

t Pelt-1 To complete the filtering equations a method for updating the

estimate of the covariance matrix of the state estimate errors as new

information becomes available is needed. The one-period ahead projection

of the covariance matrix, P, is given by:

Pefe-1 7 APe-afe-a4” + ReR’

And the optimal predictor of the covariance matrix using all the

information a time t is:

, ry-l tlt ~ Pefe-1 7 Peje-1 BY + HP. | e-12") HP e|t-1 (4)

P

In applying the above equations, several assumptions underlying the Kalman filter must be met. First, consistent estimates of the coefficient matrices A, C, R, and H, and the covariance matrices = and V are needed. These are obtained from prior estimation of the state and observation equations. Second, the model requires that the errors e. and VE be serially independent. Given the errors in the state equation are the residuals from a time-series model they should have this property. However, the errors from the observation equation represent data revisions, which for many macroeconomic time series are serially correlated. Therefore, the observation equation may need to be modi.fied to conform with the assumptions underlying the Kalman filter. This problem is addressed in Section 5. First, a description of the dat:a and estimates of time-series models of the final data and the data rev:.sions are presented.

3. Census construction of the retail sales data

Retail sales are receipts plus merchandise sold for credit: at establishments that are primarily engaged in retail trade. The latter is defined as the sale of merchandise to consumers for personal and household use. Sales exclude sales and excise taxes paid by customers

and finance charges. They do, however, include gasoline, tobacco, liquor

and other excise taxes, which are paid by the manufacturer and passed through to the retailer and the customer.

Estimates of retail sales are constructed from a sample that consists of very large businesses, which report every month, plus three panels drawn from the list sample and twelve area panels.> Based on the dollar volume of sales, businesses that report on a continuous basis comprise 35 percent of the sample, list panels that report every three months for the preliminary and final estimates account for another 59 percent, and the remaining 6 percent consists of area panels that report every twelve or six months for the preliminary and final estimates.

For each month, there are three successive estimates of retail sales -- referred to as the advance, preliminary, and final estimates. The advance estimate is published roughly ten days after the date to which the data refer and is based on a fixed subsample of the very large businesses reporting on a continuous basis, the list panels, and the area panels. The preliminary and final estimates are issued one and two months, respectively, after the advance estimate appears and are based on the fixed sample of very large businesses together with a series of rotating panels.

Each rotating panel reports two months of data at each enumeration. So, for example, the panel reporting in September gives its sales figures for July and August. In October, another panel records sales for August and September, and so on. Hence, for each month there

are two panel estimates of retail sales. The Census Bureau combines each

3. The list sample is a probability sample selected from the retail employers (SIC categories 52-59) contained in the Census Bureau’s Standard Statistical Establishment List, which effectively covers all

employers who made social security payments for their employees under FICA.

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of these panel estimates with the fixed component of the sample -- those stores reporting monthly -- to obtain what they term the "unbiased" estimates of retail sales. For any given month, t, there are two unbiased estimates, which are denoted as c+" and ceo: The preceeding subscripts denote the time the information becomes available while the succeeding subscripts denote the date to which the data pertain. Thus, t41"t is based on the panel that reports with a one month lag and the fixed subsample reporting monthly, and cee is based on the panel that reports with a two month lag in addition to the fixed subsample.

The preliminary and final estimates for each three-digit SIC level are constructed from the unbiased estimates according to the

following formulas:

u t4+1PTe, 7 25.4 + .75 t+] t ePrey_1 (5) tt1"t-1 ce2Fing 8 pypPre, + -2 pote (6)

where: pre = preliminary estimate

fin = final estimate

These filters are designed to smooth changes in unbiased estimates of retail sales due to variation across panels. Equation (5) indicates that the preliminary estimate is a weighted average of the unbiased estimate reported with a one month lag and an estimate that is obtained by multiplying the previous month's preliminary estimate by the ratio of current-to-previous month’s unbiased estimates of sales based on the

current panel of stores. This procedure attempts to put the new panel's

-9.-

data on a comparable basis with the previous panel's data. The final estimate is a weighted average of the preliminary estimate for the same

month and the new panel's unbiased estimate of sales and thus

> t+24e? averages the two panels that report sales in a given month.

The advance estimate is also constructed by applying a filter to the reported data. Recall that the advance estimate is based on a fixed subsample of the universe represented in the preliminary and final estimates. In an attempt to put the advance estimate on a comparable basis with the more comprehensive sample the Census Bureau computes it as:

ual paddy, = tt ePTee 1 (7)

t-1%4¢-1

where ate denotes the "unbiased" estimate of sales in month t based. on the (fixed) advance subsample.

a The data used in this study begin with the revision in the Cer:sus Bureau's comprehensive sample in January 1982 and extend through mid-1986. Although this period contains several revisions in the advance sample, which is redrawn every two years, these changes are not thought to significantly hamper modeling the systematic component of the data revisions. | oo | 4. Forecasting and observation models for retail sales

In this application, the state equation describes the behavior of the preliminary and final estimates for the retail control grouping of stores, defined as total retail sales excluding automotive sales, gasoline sales, and non-consumption items. Rather than model the

preliminary and final series directly, the filters used by the Census

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Bureau were inverted to recover the raw or unbiased estimates of retail sales because the systematic correlations induced by the sampling procedures were thought to be more evident in the raw data.“ That is,

equations (5) and (6) were solved to obtain:

_ pre ' 1254.75 _tPFe +4 t+1t-1

c+2"e > tceoFing - 4 44 7Pre, (6") Hereafter, the two estimates of retail sales for a given month will be

referred to as ul and u2, where ul _ uy and u2 =

t+1 t+2"t°

In modeling the time-series behavior of retail sales, nonseasonally adjusted data were used and seasonal patterns were estimated as part of the model. In addition to the usual calenda:: effects associated with monthly data, the panel rotation procedures described previously introduce a correlation at three month intervals --

corresponding to the period between successive reports by a given panel.

Furthermore, because the panels reporting every three months comprise

nearly 60 percent of the sample, a correlation at the panel rotation

frequency should be noticeable in the data. A second influence on

Set aeenweealinecenenacemmanineneenatanichsiaarmemetentsinacietenetinmmeniaed

4. Recall the Census filters are applied at a disaggregated level and then preliminary and final estimates are aggregated across categories representing major kinds of business to obtain sales for the retail control grouping of stores. However, in the procedure above, the unbiased estimates were reconstructed by applying equations (5’) and (6’) to the final and preliminary estimates for total retail control. This methodology will not reproduce exactly the original unbiased series because the filter given in equations (5) and (6) is nonlinear in the unbiased estimates.

-1ll-

seasonal patterns in retail sales is the calendar variation in the occurrence of major holidays, such as Easter. To assess these possible influences on retail sales, a basic model of the seasonality was estimated and the residuals were examined for the presence of a panel rotation and an Easter effect. The model included a constant, a time trend, a series of variables representing the composition of the month, and monthly seasonal dummies. In estimating this model the time series ul and u2 were stacked, imposing the restriction that the coefficients be the same in the two equations. However, the mean and time trend were allowed to vary across the two series.

Plots of the spectra of the residuals from this regression are shown in Figures 1 and 2. The markings on the horizontal axis indicate the seasonal frequencies for monthly data. As the monthly dummies should remove most of the power at these frequencies, the spectrum generally

takes on small values at exact multiples of x/6. The one exception

occurs at 2x/3, which is the frequency corresponding to a three-month cycle in the data. The remaining power at this frequency represents the effect of the panel rotation.

Another prominent feature of the spectral plots is that, although the power is removed at the seasonal frequencies, relatively large peaks in the spectra remain near the seasonal frequencies. This pattern suggests a seasonal influence that does not quite occur at the same interval each year. In the case of retail sales, the variable timing of Easter is probably important in accounting for the sharp peaks adjacent to the seasonal frequencies.

To examine the Easter effect, we compared the behavior of April

versus March retail sales -- after removing a deterministic trend and

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trading day effects -- with the date of Easter. A plot of the number of days between April 1 and Easter versus the difference between calendaradjusted April and March sales is given in Figure 3. Although we have relatively few observations on the timing of Easter, the relative magnitude of April versus March sales appears to be a linearly increasing function of the distance of Easter from April 1 over the interval April 1 to April 14. “Furthermore, Cleveland and Grupe (1983), in a study of this same series though with more observations, also observed a linear relationship between the date of Easter and the relative magnitude of April versus March sales.

Based on these features of the data, the following fixed effects model was specified for the two series of unbiased estimates of retail

sales:

7 log(ul) = a, TIME + ) p, TD, + 7 EASTER (8) i-1 3 3 + ) ) 5; $443; MONTH, 3, i=l j-0 7 log(u2) = @,TIME + ) p, TD, + 7 EASTER (9) ‘i=l 3 3 + ) ) 85 $4435-1 MONTH, 35.1

i=1 j=0

M8 AA BC ae BIN Ate hE

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Figure 3

EASTER EFFECT IN RETAIL CONTROL

10

U1

U2

-8 -4 0 4 8 12 16 ~=— 20 24 | DAY OF EASTER MINUS APRIL 1 “APRIL MINUS MARCH RESIDUAL SALES (BILLIONS OF DOLLARS)

- 13 -

where: TD, = trading day factors.

EASTER = a dummy variable for April that equals the number of days in a 2-week interval prior to Easter falling in March.

MONTH; 43; = monthly dummies.

The 6;'s represent the panel effects, and the ¢;'s capture the usual monthly effects. Notice that these two coefficients enter multiplicatively, so that a given month's estimate can be viewed as the interaction of a panel rotation and a monthly effect. Exploiting. the fact that a given panel reports the current month’s sales in the ul series and the previous month's sales in the u? series, the 6,'s are identified by imposing cross-equation restrictions. Because this procedure only allows us to identify relative factors for one panel versus another rather than three separate panel effects, a normalization was chosen in which one of the 6,'s was set equal to one.”

The forecasting model for retail sales consists of the fixed

effects model given by equations (8) and (9) together with a time-series

5. To see the difficulty in identifying three separate panel effects, consider unconstrained estimation of each equation. In that case, 12 monthly factors are estimated for each series. If each of these monthly factors is really the product of a panel effect and a monthly effect, then, under the hypothesis that the underlying monthly effects are the same for the two series, the panel effects can be recovered indirectly by forming ratios of the estimated monthly factors in the two equations.

Denoting the estimated monthly factors for the two series as a. 401) ard a; (2),

a, (1) _ rel a. (2) 6. 1 J

Notice that this procedure only provides estimates of the ratio of successive panel effects.

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model of the residuals. Because ul and u2 both contain estimates of retail sales for a given month, a vector autoregressive process, which includes residuals from both series was specified for the residuals. The seasonal parameters and the coefficients of the time series model were estimated simultaneously. To determine the appropriate lag length, a general model that includes lag parameters at the low frequencies as well as at the panel rotation and seasonal frequencies was tested against more

restricted alternatives. Denoting the residuals from the fixed effects

model, which are in logs, as ul and u2, the model selected was: ul - u24 = P41, (4l_y - u2_») + P19 (ul_, - u2_3) + el (10) u2 = Po1¥2_4 + Pool _o + Po342_3 + e2 (11)

Estimates for this model are displayed in Table 1. The seasonal and trading day effects all appear statistically significant and of reasonable magnitudes. December implicitly has the largest seasonal factor and the trading day coefficients generally increase approaching the end of the week with Friday and Saturday having the largest weights. The Easter effect is significant and negative as an early Easter depresses April retail sales. Recalling that the factor for panel one was normalized to 1, the relative factor for the second panel is significantly different from the factors for panels one and three at the 95 percent confidence level, while panels one and three are

insignificantly different.

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Table 1: Time Series Model of Unbiased Estimates of Retail Sales

Variable Coefficient Standard Error eee

constant 9.698 .225 time .0045 . 0003 tdl (Sun. ) .0316 .0080 td2 (Mon. ) 0422 .0072 td3 (Tues.) 0494 .0085 td4 (Wed.) .0365 .0075 td5 (Thurs. ) .0487 .0079 td6 (Fri.) .0532 0074 td7 (Sat.) .0531 0081 Easter -.0019 .0008 panel 2 .949 .0154 panel 3 1.013 .0170 January - .407 0055 February -.321 .021 March -.331 .0042 April -.280 .O11 May -.297 .0057 June -.274 0091 July - .330 0055 August -.299 0061 September -.293 .0090 October -.301 .0056 November -.201 .010 uly - u2» . ~.925 .127 ul_, - u2 4 -.672 ; . 132 u2_4 . 366 .138 ul 5 -.452 .343 u2_, .659 .327

eee (ul-u2_)) = py, (Ul_,-02_,) + py (Ul_,-u2_,) + el

u2 = Po1¥2_1 + Poottl_o + Po3¥2_5 + e2

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Table 1 (cont, )

7 where: ul = log (ul) - [e +a time + ) B,td, + y Easter i=l 3 + ) (6143; MONTH , 5, + £9943 jMONTHS 35) j=0 2 + ) 53343; MONTH,, | j=0 7 u2 = log(u2) - [e + a time + ) B,td, + y Easter i=1 3 3

2 +) ) °495435-1MONTH; 455.4 +) $545 ;HONTH,, 5, | i=2 j=0 j=0

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Turning to the time series properties of the data, given that ul and u2_4 represent the same panel of stores, equation (10) describes the behavior of the month-to-month change in the unbiased estimate of sales based on a consistent panel. According to the parameter estimates shown in Table 1, the month-to-month change displays significant negative second-order autocorrelation. Equation (11) indicates that the second estimate of sales in a given month follows a third-order autoregressive process. The inclusion of Ul at lag 2 rather than the own series residuals as well as u2 at lag 3 captures any remaining panel effects as u2 and ul» and U2, are all derived from the same panel of stores.

In modeling the early observations or the advance estimates of

retail sales, the Census filter given by equation (7) was inverted to

yield:

ua t= 22d, - ,Pre. 4 (7")

t-1

ua Notice only the ratio of the raw estimates for successive months can be recovered from the published advance estimate.

Taking logs, the relationship between the monthly change in sales based on the advance subsample and the monthly change based on the same rotating panel of stores was examined. Defining DUA, as the

revision in the estimate of the monthly percentage change in retail

sales,

DUA, = Alog(ua, ) - [log(ul, ) - log(u2._4)]

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a third-order autoregressive model, shown below, was found to adequately

describe the data revisions.

DUA, = -.267 DUA, , + .027 DUA, , - .142 DUA, , + 3 (12)

(.144) (.149) (.144)

rR? — .108

As a final diagnostic, the autocorrelation function of the residuals from the forecasting and observation models, equations (10) - (12), were examined. For all three equations the Box-Pierce Q(12) and Q(24) statistics failed to reject the hypothesis that the residuals are serially independent at the 5 percent critical level.

5. Problem respecification

Given the presence of systematic components in the data revisions, some modification in the formulation of the problem is required. Essentially, the state vector is expanded to incorporate the model of the data revisions, purging the observation equation errors of

any systematic components in the data revision Process. The state vector

is expressed as:

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According to the description of the retail sales data in Section 3 the observation at time t, y,, consists of: A log ua, Y% 7 log ul. iy

log u2, 9

Treating the actual data as the sum of the advance estimate and the data revision, the matrix H in the observation equation is defined

as:

Thus, the observation equation consists of a set of identities, implying that the observation equation errors are zero, apart from rounding. That

is,

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Ve = Hx,

The state equation remains:

x= AX, 4 + Cz. + Re, where Cz, represent the calendar and panel rotation effects. According to the forecasting and data revision models estimated in the preceeding

section the matrix A is specified as:

Pint Pn 74, Pg OO 0 0 Poy P92 Y P93 OO 0 1 0 Oo 0 0 oOo oO 0 A = 0 1 0 0 0 oO oO 0 0 0 0 1 0 Oo Oo 0 0 0 0 0 e) 11 No 13 0 0 0 0 0 1 oO 0 0 0 9O 0 0 oOo 1 0

where Ny» M9) 13 represent the coefficients from the data revision process -- equation (12).

The matrix R and the residual vector e are defined as:

FEE ON

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1 0 0

0 1 0 ert

0 0 0 ee 7 Cnt R = 0 0 0 ve

0 0 0

0 0 1

0 0 0

0 0 0

where Cie Soe 3p are the residuals from equations (10), (11), and (12), respectively.

6. Filtered estimates of retail sales

Using the method outlined above, filtered estimates of the unbiased figures for retail sales -- ul and u2 -- were calculated over the period May 1982 to May 1986. Recalling that ul and u2 are available with one and two period lags respectively, ul, u2., and ude 4 were estimated based on the information available at time t. Estimates of the preliminary and final series were then reconstructed from the filtered data using equations (5) and (6).

An analogous procedure is used to obtain estimates of preliminary and final retail sales figures based solely on the time series model described in equations (10) and (11). First, model

forecasts of the unbiased estimates were calculated using the same

information as before. Thus, projected values of u2i 4 were used in

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forming estimates of ul. and u2,.° .Next, these estimates were recombined according to equations (5) and (6) to obtain model-based projections of preliminary and final retail sales figures.

The performance of the filtered estimates is compared with that of the model forecast and the Census Bureau's early published data -labelled the raw observation -- in Table 2. The basis for comparison is the mean-squared error. The results show that the filtered estimates yield about a 30 to 35 percent reduction in the mean-squared error of the observation, depending on which revision is considered. In projecting economic activity, one is most concerned with an estimate of the monthto-month change in retail sales -- column 4 of the table.

The size of the gains achieved by the filter depends on the quality of the forecasts from the alternative information source -- the time series model of retail sales. If these projections are very poor relative to the quality of the raw observation, then little improvement is likely to be achieved through the use of the filter. The forecast errors of the model are of roughly the same order of magnitude as the raw observation errors, as indicated by a comparison of the mean-squared ercors in Table 2. Thus, the filter combines two independent forecasts of roughly comparable quality and thereby yields considerable improvement

in the quality of the estimates of retail sales activity.

6. The estimates of ul, and u2, formed in this way differ from the oneperiod ahead model-based projection used in the filter -- Xe | 1: The latter uses last period's filtered estimates of U2. rather! fhan the model forecast in forming estimates of ul, and u2,

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Table 2: Results of Kalman filtering of retail sales estimates

adv to adv to pre to adv-pre 4 to MSE x 10° final pre final pre-fin , 1. Raw observation 3.76 2.52 .76 1.64 2. Model projection 3.18 2.93 .63 2.07 3. State estimate 2.48 1.75 .49 1.07

4. Percent reduction from raw observation to state estimate 34 31 36 35

7. Conclusion

If forecasts of economic activity are to rely on preliminary data, the predictable component of the data revisions should be taken into account. For some economic time series, the data revisions are not forecastable based either on their own past or on other information. In these instances, the raw preliminary data provides an efficient forecast of that variable.

This study, however, suggests that revisions to retail sales are more appropriately characterized as classical errors in variables. The results indicate that substantial improvement in the information content of the early data can be obtained by combining it with a time series model of underlying retail sales activity via the Kalman filter. For the monthly change in retail sales -- the most relevant concept for forecasting economic activity -- the sum of squared errors in the early data was reduced by over 30 percent.

These results contrast with work by Mankiw and Shapiro examining the properties of revisions to GNP data. As personal consumption expenditures for goods constitute roughly 1/3 of GNP, either the systematic nature of revisions to retail sales is taken into account in constructing quarterly data on personal consumption expenditures or revisions to other components of GNP behave in such a way as to offset this correlation. |

Moreover, the Kalman filter likely can be used to improve the information content of other bodies of provisional data. In assessing

its performance, however, one caveat is in order. Although the

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improvement in forecast accuracy presented here is considerable, it is potentially overstated because the data that is being forecast was used to fit the parameters of the model -- that is, the forecast comparisons are made in-sample. A better gauge of the forecasting ability of the filter would be an examination of its out-of-sample performance, but this was not feasible because the amount of data available on a consistent

basis is quite small.

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References

Cleveland, W. P., and M. R. Grupe, "Modeling Time Series When Calendar

Effects are Present," Applied Time Series Analysis of Economic Data, ed. A. Zellner, 1983, pp. 57-67.

Cole, R., Errors in Provisional Estimates of Gross National Product, New York: National Bureau of Economic Research, 1969.

Conrad, W. and C. Corrado, "The Application of the Kalman Filter to Revisions in Monthly Retail Sales Estimates," Journal of Economic Dynamics and Control, vol. 1, pp. 177-198.

Harvey, A. C., Time Series Models, New York: Wiley, 1981.

, "The Kalman Filter and Its Applications in Econometrics and Time Series Analysis," Mimeo, London School of Economics, 1981.

Howrey, E. P., "The Use of Preliminary Data in Econometric Forecasting," Review of Economics and Statistics, vol. 60 (May 1978), pp. 193-200.

Mankiw, N. G., and M. D. Shapiro, "News or Noise? Revisions of the Preliminary Gross National Product Data," Harvard University Working Paper #1228, April 1986.

U.S. Bureau of the Census, Advance Monthly Retail Sales, various releases, (Washington, D. C.)

, 1982, Revised Monthly Retail Sales and Inventories: January 1972-December 1981 (Washington, D. C.) :

Zellner, A., "A Statistical Analysis of Provisional Estimates of Gross National Product and Its Components, of Selected National Income Components, and of Personal Saving, Journal of the American

Statistical Association, vol: 53 (1958), pp. 54-65..:

IFDP NUMBER

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- 24 - International Finance Discussion Papers

TITLES 1987

Improving the Forecast Accuracy of Provisional Data: An Application of the Kalman Filter to Retail Sales Estimates

Monte Carlo Methodology and the Finite Sample Properties of Statistics for Testing Nested and Non-Nested Hypotheses

The U.S. External Deficit: Its Causes and Persistence

Debt Conversions: Economic Issues for Heavily Indebted Developing Countries

Exchange Rate Regimes and Macroeconomic Stabilization in a Developing Country

Monetary Policy in Taiwan, China

The Pricing of Forward Exchange Rates Realignment of the Yen-Dollar Exchange Rate: Aspects of the Adjustment Process in. Japan

The Effect of Multilateral Trade Clearinghouses on the Demand for

International Reserves

Protection and Retaliatien: _ Changing the Rules of the Game

International Duopoly with Tariffs A Simple Simulation Model of International Bank Lending

A Reassessment of Measures of the Dollar’s Effective Exchange Value

Macroeconomic Instability of the Less Developed Country Economy when Bank Credit is Rationed

AUTHOR(s)

B. Dianne Pauls

Neil R. Ericsson

Peter Hooper

Catherine I.. Mann

Lewis S. Alexander

David H. Howard

Robert F. Emery Ross Levine

Bonnie E. Loopesko Robert E. Johnson

Ellen E. Meade

Catherine L.. Mann

Erie O’N. Fisher Charles A. Wilson

Henry S. Terrell Robert S. Dohner

B. Dianne Pauls

William L. Helkie

David F. Spigelman

Please address requests for copies to International Finance Discussion

Papers,

Division of International Finance, Federal Reserve System, Washington, D.C.

20551.

Stop 24, Board of Governors of the

IFDP

NUMBIER

304

303

302

301

300

298 297

296

295

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293

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291

- 25 -

International Finance Discussion Papers

TITLES

The U.S. External Deficit in the 1980s: An Empirical Analysis

An Analogue Model of Phase-Averaging Procedures

A Model of Exchange Rate Pass-Through

_ The Out-of-Sample Forecasting Performance

of Exchange Rate Models When Coefficients are Allowed to Change

Financial Concentration and Development: An Empirical Analysis of the Venezuelan Case

Deposit Insurance Assessments on Deposits at Foreign Branches of U.S. Banks

1986

The International Debt Situation

The Cost Competitiveness of the Europaper Market .

Germany and the European Disease

The United States International Asset and Liability Position: A Comparison of Flow of Funds and Commerce Department

An International Arbitrage Pricing Model with PPP Deviations

The Structure and Properties of the FRB Multicountry Model

Short-term and Long-term Expectations of the Yen/Dollar Exchange Rate: Evidence from Survey Data

Anticipated Fiscal Contraction: The Economic Consequences of the Announcement of Gramm-Rudman-Hollings

AUTHOR (s)

William L. Helkie Peter Hooper

Julia Campos

Neil R. Ericsson David F. Hendry Eric O'N. Fisher Garry J. Schinasi

P.A.V.B. Swamy

Jaime Marquez Janice Shack-Marquez

Jeffrey C. Marquardt

Edwin M. Truman Rodney H. Mills

John Davis

Patrick Minford Guido E. van der Ven John E. Wilson

Ross Levine

Hali J. Edison Jaime R. Marquez Ralph W. Tryon

Jeffrey A. Frankel Kenneth A. Froot

Robert A. Johnson