The geometric distributed lag model, after application
of the so-called Koyck transformation, is often used to establish
the dynamic link between 

Franses, Ph.H.B.F. (Philip Hans)

Oest, R.D. (Rutger) van

Franses, Philip Hans

van Oest, RD (Rutger)

EUR Research Repository

On the econometrics of the Koyck model

The geometric distributed lag model, after application
of the so-called Koyck transformation, is often used to establish
the dynamic link between sales and advertising. This year, the
Koyck model celebrates its 50th anniversary.In this paper we focus 
on the econometrics of this popular model,and we show that this 
seemingly simple model is a little more complicated than we always 
tend to think. First, the Koyck transformation entails a parameter 
restriction, which should not be overlooked for efficiency reasons. 
Second, the t-statistic for the parameter for direct advertising effects has a non-standard distribution. We provide solutions to these two issues.
For the monthly Lydia Pinkham data, it is shown that various
practical decisions lead to very different conclusions

Erasmus University Digital Repository

On the econometrics of the Koyck model∗Philip Hans Franses and Rutger van OestEconometric Institute, Erasmus University RotterdamP.O. Box 1738, NL-3000 DR, Rotterdam, The NetherlandsEconometric Institute Report 2004-07AbstractThe geometric distributed lag model, after application of the so-called Koycktransformation, is often used to establish the dynamic link between sales andadvertising. This year, the Koyck model celebrates its 50th anniversary.In this paper we focus on the econometrics of this popular model, and weshow that this seemingly simple model is a little more complicated than wealways tend to think. First, the Koyck transformation entails a parameterrestriction, which should not be overlooked for efficiency reasons. Second, thet-statistic for the parameter for direct advertising effects has a non-standarddistribution. We provide solutions to these two issues.For the monthly Lydia Pinkham data, it is shown that various practicaldecisions lead to very different conclusions.Key words and phrases: Koyck model, sales-advertising relationship∗We thank Dennis Fok, Richard Paap and Marco Vriens for helpful discussions, and LotjeKruithof for excellent research assistance. The address for correspondence is Econometric Institute,Erasmus University Rotterdam, P.O. Box 1738, NL-3000 DR Rotterdam, email: franses@few.eur.nlor vanoest@few.eur.nl. The computer programs used in this paper are available from the secondauthor.11 IntroductionThe geometric distributed lag model is often used to investigate the current andcarryover effect of advertising on sales. This model makes current sales a function ofcurrent and past advertising levels, where the lag coefficients have a geometricallydecaying pattern. As this model involves an infinite number of lagged variables, oneoften considers the so-called Koyck transformation (Koyck, 1954). In many studiesthe resultant model is hence called the Koyck model. Leendert Marinus Koyck (1918-1962) was a Dutch economist who studied and worked at the Netherlands School ofEconomics, which is now called the Erasmus University Rotterdam.In this research note we will discuss the basic Koyck model, and illustrate thatthis model is less straightforward to analyze than is usually assumed or suggested.We will provide a discussion of possible solutions. Next, we will show for the well-known monthly Lydia Pinkham data that various approaches lead to different con-clusions, thereby emphasizing the relevance of the proper methods.2 The Koyck modelConsider the variables sales St and advertising At, and assume that the link betweenthese two variables is given bySt = µ+ β(At + λAt−1 + λ2At−2 + ...) + εt, (1)where εt is an uncorrelated error variable. The parameter λ is usually called theretention rate. The current effect of advertising is β, whereas the carryover effect isequal to β1−λ . To provide some intuition for the discussion below, note that whenβ = 0, the parameter λ disappears from the model, and cannot be retrieved. From amarketing perspective this is no problem as the model assumes that the current andcarryover effects of advertising are both a function of β. However, from a statisticalperspective, matters become a little complicated.As this model contains a very large number of variables, it is common practiceto apply the Koyck transformation. This entails subtracting λSt−1 from (1), to getSt = µ+ βAt + λSt−1 + εt − λεt−1. (2)1In time series jargon, this model is called an ARMAX model, see Franses (1991)for more details on ARMAX models. The autoregressive [AR] part concerns St−1,the moving average part [MA] concerns εt−1 and the explanatory variables part [X]concerns At. Note that the parameter λ appears twice, and hence that, except forthe intercept, there are only two parameters to estimate, while the model effectivelycontains three explanatory variables. Additionally, note that when β = 0, the modelcontains the lag polynomial 1 − λL, with L the lag operator, on both sides, whichgets cancelled, that is, when β = 0, the model reduces to St = µ+ εt.Parameter estimationThere are several approaches that one can follow to estimate the parameters in theresulting Koyck model in (2). Of course, the appropriate estimation method hereis the maximum likelihood method, which imposes that the AR and MA parameterare the same. The (conditional) log-likelihood function is given bylnL(µ, β, λ, σ2) = − T − 12(ln(2pi) + ln(σ2))−T∑t=2ε2t2σ2, (3)where T denotes the number of observations, and the {εt} are recursively defined asε1 = 0,εt = St − µ− βAt − λSt−1 + λεt−1, t = 2, . . . , T. (4)This approach is also described in Hamilton (1994, p. 132) for general ARMAmodels. Asymptotic standard errors are obtained by taking the square roots of thediagonal elements of the estimated covariance matrix, which in turn can be computedas minus the inverse of the Hessian of (3) evaluated for the optimal parametervalues. Numerical techniques, such as the BFGS algorithm or the Newton-Raphsonalgorithm, have to be used to get the maximum likelihood parameter estimates.It is tempting though to decide to all the way neglect the MA part, so that themodel parameters can be estimated using the method of ordinary least squares. Thisis obviously not a sound approach, as εt−1 and St−1 are not uncorrelated, and therebyone of the basic premises of regression theory gets violated. Simulation results inTable 1 suggest that the resultant downward bias of λˆ can be substantial.2One can also decide to estimate the parameters in an unrestricted ARMAXmodel, that is,St = µ+ βAt + λ1St−1 + εt − λ2εt−1. (5)It is possible to get estimators for λ1 and for λ2 which are consistent estimatorsfor λ. In practice, however, it is most likely that the corresponding estimates takedifferent values, and the question appears which one should take. Also, (5) can-not be transformed back to a model like (1), and hence is less interesting from atheoretical perspective. Moreover, correctly imposing that λ1 = λ2 yields a moreefficient estimator. The simulation results in Table 1 suggest that estimating λ1 inan unrestricted ARMAX model leads to almost no bias.Testing advertising effectsA second issue of concern for the Koyck model is a test for the significance of theadvertising effects. Indeed, one may want to examine whether β is equal to zero. Thisis not trivial as under the null hypothesis of interest, that is, β = 0, the parameterλ disappears from the model, see (1) and (2). This is what is known as the Davies(1977) problem, and it seriously complicates statistical analysis. The issue is thatthe usually considered t-statistic depends on λ for which it is not clear which value totake. An appealing approach might seem to simply set λ at its maximum likelihoodvalue. However, this would make the test (and its critical values) dependent on thedata, and the asymptotic distribution would not be standard normal.Recent solutions to the Davies problem are provided by Andrews and Ploberger(1994) and Hansen (1996), see also Carrasco (2002). The main idea is that oneconstructs a new test statistic based on the entire distribution of the original teststatistic over a range of values of the unidentified parameter λ. In the Koyck model,a sensible range for λ would be the interval [ 0, 1). One possibility, involving theentire distribution over λ, would be to consider the class of “sup test statistics”,which corresponds to the highest value of the original test statistic within the rangefor λ. This approach is advocated by Davies (1977), see also Hansen (1996) andCarrasco (2002). Alternatively, one can consider the class of “ave test statistics”,based on the average value of the original test statistic. This approach is put forward3by Andrews and Ploberger (1994), and is further investigated by Hansen (1996). Ineach case, the asymptotic distribution of the resulting test statistic is not standardnormal, so that its distribution has to be simulated.In this paper we consider the “ave” and “sup” versions of the absolute t statistic|tβ| and the Wald statistic t2β, where tβ is obtained by taking the ratio of the maxi-mum likelihood estimate of β and its estimated asymptotic standard error. So, wefocus on four test statistics, that is, ave absolute t, ave Wald, sup absolute t, andsup Wald. Although the two sup tests are equivalent, we include them both in orderto achieve symmetry. Table 2 contains the simulated (asymptotic) critical values forthe four tests at confidence levels of 80%, 90%, 95%, and 99%. In order to obtainthese critical values, we ran 40000 simulations for T = 1000 observations. In eachsimulation, advertising data were drawn from a standard normal distribution. Next,sales data were generated from the Koyck model under the null hypothesis β = 0,that is, we set β = 0, µ = 0, and we assumed variance 0.25, like in Table 1. Foreach simulation, the four test statistics were computed, and their values were stored.Finally, the four resulting samples were ordered in an ascending way, so that thequantiles became available. In each simulation, the sup absolute t and sup Waldstatistics were obtained via a grid search over λ from 0 to 0.999 with step size 0.001.Before we turn to an application of these tests to real-life data, it seems wise tosee which of these tests performs best in practice. For that purpose, we ran anotherset of simulations, and the results are given in Table 3. Clearly, the power of thesupremum tests is not very high, and hence we would recommend the use of theaverage tests.3 An illustrationOne might now be tempted to think (or hope) that the above considerations wouldnot have a substantial impact on empirical analysis. Unfortunately, they do, as canalready be illustrated for the illustrious monthly Lydia Pinkham data, which havebeen used in many marketing studies.4If one decides to neglect the MA part of the Koyck model, that is,St = µ+ βAt + λSt−1 + vt, (6)and replaces the intercept by twelve monthly dummies to correct for seasonal effects,then β gets estimated at a value of 0.360, with standard error 0.118, and λ isestimated equal to 0.370, with standard error 0.125.Unrestricted estimation of the Koyck model, that is, (5) with monthly dummies,renders a β of 0.335 (0.101), a λ1 of 0.690 with standard error 0.119, and a λ2 of0.561 with error 0.178. Clearly, these two λ parameters are rather different. Noticethat the t−ratios for β in the above two models are 3.039 and 3.307, respectively.The maximum likelihood estimates of β and λ for (2), again including monthlydummies, are found to be 0.339 (0.098) and 0.703 (0.125) respectively, where thereported standard errors are asymptotic standard errors. Consistent with the simu-lation evidence from Table 1, the estimates of λ1 in the unrestricted model and λ inthe restricted Koyck model are approximately equal (0.690 and 0.703). Furthermore,it can be seen that ignoring the MA part of the Koyck model indeed results in seriousunderestimation of the retention rate (0.370 versus 0.703), as suggested by Table 1.To put it differently, neglecting the MA component would result in a 90% durationinterval of 1.3 months, whereas appropriate maximum likelihood estimation wouldresult in a much longer 90% duration interval of 5.5 months1.The values of the four (average and supremum) statistics for testing β = 0 are4.61, 22.06, 5.55 and 30.79, respectively. By comparing these realized values with thecritical values in Table 2, we conclude that the two “ave tests” indicate a significantadvertising effect at a 1% significance level, whereas the two “sup tests” fail to rejectthe null hypothesis at a 20% level. These contradictory results confirm our findingsin Table 3 which suggests that the supremum tests do not have much power.Finally, for illustrative purposes, Figure 1 shows the underlying distributions ofthe absolute t statistic |tβ| and the Wald statistic t2β over the different values of λ.1Clarke (1976, p346) defines the (100 × α)% duration interval as the time period τα duringwhich (100 × α)% of the expected cumulative advertising effect has taken place. It can be shownthat τα =ln(1−α)ln(λ) − 1.5For both statistics, the supremum corresponds to λ = 0.663, which is quite close tothe maximum likelihood estimate λˆ = 0.703.4 ConclusionThe Koyck model is often applied in marketing practice, but it is more complicatedto analyze than one would perhaps think. Proper parameter estimation requiresimposing parameter restrictions in the estimation routine. And, proper inferenceon the advertising effects requires new test statistics with non-standard asymptoticdistributions, as the retention parameter disappears under the null hypothesis ofno effect of advertising. In this paper we showed how these tools work. For themonthly Lydia Pinkham data we showed that various tests can lead to contrastingconclusions.6Table 1: Estimating the retention parameter λ in a Koyckmodel. In case (A) the MA term is neglected. In case (B) themodel parameters are estimated using nonlinear least squareswith unrestricted parameters λ1 and λ2, where only λ1 is re-ported. The simulation results are based on 1000 replications.The cells contain the mean value of the 1000 estimates andthe associated standard deviation. Advertising data are drawnfrom a standard normal distribution. Next, sales data are gen-erated for µ = 0, β = 1 and an error process having variance0.25.n = 50 n = 500 n = 5000λ = 0.5A 0.412 (0.062) 0.420 (0.018) 0.420 (0.012)B 0.502 (0.053) 0.500 (0.014) 0.500 (0.010)λ = 0.8A 0.705 (0.053) 0.732 (0.013) 0.734 (0.004)B 0.797 (0.025) 0.800 (0.005) 0.800 (0.002)λ = 0.9A 0.825 (0.046) 0.855 (0.010) 0.859 (0.003)B 0.899 (0.014) 0.900 (0.002) 0.900 (0.001)λ = 0.95A 0.879 (0.051) 0.923 (0.008) 0.927 (0.002)B 0.948 (0.012) 0.950 (0.008) 0.950 (0.000)7Table 2: Critical values of various tests for the hypothesis that β = 0 in the Koyckmodel. Number of replications is 40000. The sample size is 1000. The grid for λ runsfrom 0.000 to 0.999 with step size 0.001. Data are generated similar to those in Table1.Confidence level Ave absolute t Ave Wald Sup absolute t Sup Wald80 % 1.21 1.67 6.18 38.1390 % 1.53 2.56 7.47 55.8695 % 1.80 3.50 8.41 70.6599 % 2.34 5.79 9.80 95.998Table 3: Empirical power of various tests for the hypothesis that β = 0in the Koyck model. Number of replications is 1000 for each value of β.The sample size is 1000. The critical value is set at the 5% level. Data aregenerated similar to those in Table 1.β Ave absolute t Ave Wald Sup absolute t Sup Wald0.00 5.90 5.70 5.20 5.200.02 29.80 29.40 5.30 5.300.04 79.30 78.70 4.50 4.500.06 98.20 98.20 5.30 5.300.08 100.00 100.00 7.40 7.400.10 100.00 100.00 16.80 16.8090.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.02345absolute t × lambda 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0102030 Wald × lambda Figure 1: Testing for the significance of advertising effects in the Koyck modelapplied to the monthly Lydia Pinkham data. The values of the absolute t statisticand the Wald statistic are shown for different values of the retention parameter λ.10ReferencesAndrews, D.W.K. and W. Ploberger (1994), Optimal tests when a nuisance param-eter is present only under the alternative, Econometrica, 62, 1383-1414.Carrasco, M. (2002), Misspecified structural change, threshold, and Markov-switchingModels, Journal of Econometrics, 109, 239 - 273.Clarke D.G. (1976), Econometric measurement of the duration of advertising ef-fect on sales, Journal of Marketing Research, 8 (November), 345 - 357.Davies, R.B. (1987), Hypothesis testing when a nuisance parameter is present onlyunder the alternative, Biometrika, 64, 247-254.Franses, P.H. (1991), Primary demand for beer in The Netherlands: An applica-tion of ARMAX model specification, Journal of Marketing Research, 28, 240-245.Hamilton, J.D. (1994), Time Series Analysis, Princeton University Press.Hansen, B.E. (1996), Inference when a nuisance parameter is not identified underthe null hypothesis, Econometrica, 64, 413-430.Koyck, L.M. (1954), Distributed Lags and Investment Analysis, Amsterdam: North-Holland.11

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