A UNIFYING VIEW ON MULTI-STEP FORECASTING USING AN AUTOREGRESSION

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1 doi: /j x A UNIFYING VIEW ON MULTI-STEP FORECASTING USING AN AUTOREGRESSION Philip Hans Franses and Rianne Legerstee Econometric Institute and Tinbergen Institute, Erasmus University Rotterdam Abstract. This paper unifies two methodologies for multi-step forecasting from autoregressive time series models. The first is covered in most of the traditional time series literature and it uses short-horizon forecasts to compute longer-horizon forecasts, while the estimation method minimizes one-step-ahead forecast errors. The second methodology considers direct multi-step estimation and forecasting. In this paper, we show that both approaches are special (boundary) cases of a technique called partial least squares (PLS) when this technique is applied to an autoregression. We outline this methodology and show how it unifies the other two. We also illustrate the practical relevance of the resultant PLS autoregression for 17 quarterly, seasonally adjusted, industrial production series. Our main findings are that both boundary models can be improved by including factors indicated from the PLS technique. Keywords. Autoregression; Multi-step forecasting; Partial least squares 1. Introduction This paper deals with multi-step forecasting from an autoregressive time series model. Such a model (and its variations) is very often applied in practice, where applications range from finance, marketing, macroeconomics, international economics and others. Multi-step macroeconomic forecasts are perhaps most well known as newspapers regularly quote such forecasts that cover the next five to 10 years for economies in developed and developing countries. Most well known are perhaps the multi-step forecasts that are made by institutes such as the World Bank and the International Monetary Fund for many national accounts variables. At present there are two commonly applied methods to create multi-step forecasts from an autoregression. The first is what Chevillon (2007) in a recent survey in this journal calls the iterated multi-step (IMS) method. This method entails that to forecast, say, two steps ahead one imputes the one-step forecast. The parameters in the autoregression are usually estimated using ordinary least squares (OLS) (or a variant) and for the IMS method this means that the estimation method seeks to minimize the squares of the residuals, which equal the one-step-ahead forecast, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.

2 390 FRANSES AND LEGERSTEE errors. The second commonly applied method, which is also surveyed in Chevillon (2007), is called the direct multi-step (DMS) method, and this entails that a forecast for each multi-step horizon is created from an autoregression where each time the parameters are estimated anew, where now not the squared one-step-ahead errors but in fact the associated squared multi-step-ahead errors are minimized. The IMS method is most often advocated in standard time series textbooks, such as for example Box et al. (1994), Brockwell and Davis (2002), Franses (1998) and Hamilton (1994). One reason might be that estimation of the parameters is rather straightforward in this IMS case. Also, (recursive) forecast formulas for multi-step forecasts are easy to derive, and using these it is also relatively easy to derive confidence bounds around these forecasts. On the other hand, the DMS method is studied in detail by Tiao and Xu (1993), Weiss (1991), Bhansali (1996) and Kang (2003) among others, where these studies derive properties of OLS-based estimators, propose model selection criteria and give illustrations of the merits of this approach. A recent empirical comparison of IMS and DMS methods is given in Marcellino et al. (2006). A recent survey of the DMS method is given in Chevillon (2007). One of the important features of the DMS method, that is also highlighted in that survey paper, is that the DMS method is asymptotically more efficient, i.e. smaller confidence bounds are to be expected around multistep forecasts, as compared with the IMS method, which is of course due to the estimation method itself. On the other hand, Chevillon (2007) concludes that when the data are stationary and the model is well specified, there will not be substantial differences in forecast accuracy across the IMS and DMS methods. Only when the model may be misspecified is some gain to be expected for the DMS method over the IMS method. 1.1 Outline of Paper In the present paper, we will show that the IMS and DMS methods can be viewed as boundary cases of a more general method of forecasting for the one-step and multi-step horizons using an autoregression. This general method is based on the so-called partial least squares (PLS) technique, which in essence is the regression extension of the familiar canonical correlations (CCA) technique. When applying PLS to an autoregression, one assumes that past observations are used to forecast one-step to multi-step observations all at the same time. The PLS method is often used in cases where the number of explanatory variables is too large relative to the sample size, which may happen in survey research in various social sciences; see for example Heij et al. (2008) and Stock and Watson (2002) who use the related principal components analysis (PCA) technique. But we will show that the basic idea of PLS can equally well be applied to forecasting from an autoregression. The space between the boundary IMS and DMS cases is created by the fact that the DMS method does not incorporate that the forecasts for the various horizons are most likely to be correlated. This is in contrast with the IMS method which is fully based on those correlations. The intermediate cases, in a sense, allow for varying degrees of correlation, and hence IMS and DMS take extreme positions.

3 UNIFYING VIEW ON MULTI-STEP FORECASTING USING AUTOREGRESSION 391 Note that jointly estimating models for several horizons at once is also the topic of Liu (1996) and Weiss (1991), but the PLS approach is quite different. The outline of the paper is as follows. In the next section we reiterate the above, but now using more formal expressions. Section 3 describes the PLS technique and introduces the unifying model to be called a partial least squares autoregression (PLSAR). In Section 4, the actual use of the PLSAR model is illustrated first for a lengthy quarterly industrial production series, here the USA, and second for 16 such series for other countries. The remarkable outcome of the forecast evaluations is that the IMS is about best in 20% 30% of the cases, that the DMS method rarely is best, while in 60% 70% of the cases the IMS method can be improved by including PLS features. This suggests that this unifying PLS methodology opens ways for improvement in various practical settings. The final section therefore outlines a variety of further research topics. 2. A Unifying View This paper deals with an autoregressive univariate time series model of order p for a time series y t, t = 1, 2,...,n, that is to be used for forecasting 1 to h steps ahead, where the focus is only on point forecasts. An extension to distributed lag models and multiple time series should not be too complicated, but will not be dealt with in the current paper. Also, the derivation of confidence bounds around these point forecasts is postponed to future work. 2.1 A Single Model for all Horizons (IMS) The autoregressive model of order p (AR(p)), i.e. y t = μ + ρ 1 y t 1 + ρ 2 y t 2 + +ρ p y t p + ε t (1) is often used for forecasting. In practice the parameters are usually estimated using OLS, which here means that the sum of squared one-step-ahead forecast errors is minimized. The one-step-ahead forecast at time n is generated as ŷ n+1 = ˆμ + ˆρ 1 y n + ˆρ 2 y n ˆρ p y n (p 1) (2) The two-step-ahead forecast is created as ŷ n+2 = ˆμ + ˆρ 1 ŷ n+1 + ˆρ 2 y n + + ˆρ p y n (p 2) (3) and so on, while finally, when h > p, theh-step forecast follows from the recursion ŷ n+h = ˆμ + ˆρ 1 ŷ n+h 1 + ˆρ 2 ŷ n+h 2 + ˆρ p ŷ n (p h) (4) In this IMS case, a single time series model is used for all forecast horizons. This method of substituting earlier forecasts to forecast further ahead implies that, for h large, the forecast values converge in probability to the unconditional mean, i.e μ ŷ n+h (5) 1 ρ 1 ρ 2 ρ p

4 392 FRANSES AND LEGERSTEE This expression immediately indicates that a drawback of the IMS method is that multi-step forecasts converge to the unconditional mean, and, depending on the value of the parameters, this convergence can happen rather fast. 2.2 For Each Horizon a Different Model (DMS) There are various reasons why an alternative strategy for forecasting 1 to h steps ahead is better. The first is that OLS applied to equation (1) aims at minimizing the sum of squared ˆε t, which amounts to the sum of squared one-step-ahead forecast errors. Indeed, abstaining from estimation errors, the difference between equations (1) and (2) is ε n+1. There is no guarantee, however, that this minimization also implies a minimum of the sum of squared h-step forecast errors. The second reason is that one may simply want to have different models for different forecast horizons, i.e. for each h one would want to have a different model. The third reason is that for stationary time series h-step forecasts quickly converge to the unconditional mean as in equation (5), which may be implausible in some practical situations. An obvious alternative is then to have different models for different forecast horizons, and hence to replace equation (1) by y t+h = μ h + ρ 1,h y t + ρ 2,h y t 1 + +ρ p,h y t (p 1) + ε t,h (6) for h = 1, 2,..., where the variance of ε t,h is σh 2 and hence can also vary with the forecast horizon. This notion of having a different forecasting model for different forecasting horizons comes close to the idea behind periodic models, which allow for different models for different seasons (see Franses and Paap (2004) for a recent review). The line of thought in equation (6) is followed in Tiao and Xu (1993), Weiss (1991), Bhansali (1996) and Kang (2003) among others, where these studies derive properties of OLS-based estimators, propose model selection criteria and give illustrations of the merits of this approach (see Chevillon (2007) for an excellent recent survey). Note that the model orders may also differ across horizons, so one may also use the notation p h instead of p. For ease of exposition, this extension is not considered here. 2.3 Towards a Unifying Model To have a different model for different forecast horizons has an important shortcoming and that is that in these models the correlation between adjacent time series observations is not exploited. Indeed, if one believes in the relevance of an autoregression, then one implicitly assumes that the variables y t and y t j are correlated. If this is the case, then also the forecasts y n+h and y n+h j for any j must be correlated. The IMS method above fully exploits this correlation, and in fact its forecasts are fully determined by it. In contrast, the DMS method fully neglects this correlation. In fact, it may occur that multi-step forecasts, when lined up as separate time series, have different correlation structures than the in-sample data are assumed to have when the autoregression is well specified.

5 UNIFYING VIEW ON MULTI-STEP FORECASTING USING AUTOREGRESSION 393 To provide a unifying view on the creation of multi-step forecasts, one would allow for intermediate degrees of correlation across these forecasts. One way to do so is to think of forecasting from an AR(p) model by jointly predicting (y t+h, y t+h 1, y t+h 2,...,y t+1 ) from (y t, y t 1,...,y t (p 1) ) This notion can be recognized as an extension of the familiar CCA technique, and the regression technique that enables one to do this is called PLS. It is this method that will be introduced in this paper for jointly forecasting 1 to h steps ahead from an AR(p) model. It will be shown that the resulting PLS autoregression (PLSAR) amounts to cases in between equations (1) and (6). In a sense, one can view PLS as some strong form of seemingly unrelated regressions with correlated errors. The outline of the rest of this paper is as follows. In Section 3 the PLS autoregressive model for order (h, p) is introduced. Parameter estimation and forecasting will be addressed as well. Section 4 illustrates in detail this new model for quarterly seasonally adjusted US industrial production. Next, the empirical analysis is extended to cover 16 more such series. Section 5 contains various topics for further research. 3. PLS Autoregression The interest is in jointly predicting (y t+h, y t+h 1, y t+h 2,...,y t+1 ) to be collected in Y for notational convenience, using the available information on the time series until and including t, i.e. for (y t, y t 1,...,y t (p 1) ), which will be collected in X. Note that the focus is on prediction and not on correlation. The latter would amount to computing the so-called CCA; see Esposito Vinzi et al. (2007) for details of the PLS technique and its comparison with related methods. 1 Our paper is the first to address the use of PLS for multi-step forecasting. 3.1 Representation The data matrices that are the input to the regression problem are Y and X, andthey are of size n h and n p, respectively. 2 The idea behind PLS is that it seeks components of X which are also relevant for Y. PLS regression aims at finding a set of latent variables which together simultaneously decompose Y and X, given that these latent variables explain most of the covariance between Y and X. Note that this last feature makes PLS different from PCA. Suppose that there are k such latent variables, where k can take values from 1 to p. The first step is now to decompose X as X = KW (7) where K collects these k latent variables, with K of size n k, and where the loadings are collected in a k p matrix W.

6 394 FRANSES AND LEGERSTEE Next, it is assumed that the fit of Y, given the PLS regression model, is Ŷ = KBC (8) Here the regression weights are collected in the k k matrix B and C is again a loading matrix, where now these loadings concern K on Ŷ.ThisC matrix is of size k h, and it plays the same role as factor loadings in PCA. These two equations underlie the PLS regression model, which in the present application is called a PLSAR of order (h, p), or PLSAR(h, p). 3.2 Estimation The method of estimation of the components in K, the parameters in B and the loadings in W and C bears similarities with related techniques such as PCA and CCA (see also Esposito Vinzi et al. (2007) and for example Abdi (2003)). A popular approach is an iterative least squares method (see Abdi (2003) for the sequential steps) which is also used below in the empirical section. Another approach would follow eigenvalue and singular value decompositions, and in practice small numerical differences can be expected. Note that, as is usual, prior to estimation the variables in Y and X are scaled towards z-scores (i.e. the mean is subtracted and the variables are scaled with their standard deviation) in order to facilitate computations. Like PCA, the estimation method also delivers the degree of variance of Y and that of X that is explained by each of the latent variables in K. Comparing the percentages of explained variance allows one to fix the value of k, if one indeed wantstomakeachoice. 3.3 Forecasting Taking equations (7) and (8) together shows that the forecasting scheme based on the PLSAR is Ŷ = X ˆB pls (9) where ˆB pls is a p h matrix computed as ˆB pls = Ŵ 1 ˆBĈ, where the inverse typically is taken as the Moore Penrose inverse. In the present paper the focus is on the quality of the forecasts Ŷ, which of course are ŷ n+h, ŷ n+h 1, ŷ n+h 2,..., ŷ n The Unifying View When K is a full rank matrix, i.e. when k is equal to p, then the number of latent variables is equal to the number of explanatory variables. In that case the ˆB pls matrix has full rank too, and hence equation (9) implies a different model for each of the columns of Y. In the case of the autoregression this then approximately boils down to the DMS method, i.e. a model like equation (6). At the other end, when ˆB pls has rank 1, then it is easy to see that this means that each forecast horizon is

7 UNIFYING VIEW ON MULTI-STEP FORECASTING USING AUTOREGRESSION 395 related to the same set of parameters for X, which in the case of an autoregression means that there is a single AR(p) model like equation (1), i.e. the IMS method. Note that in practice, due to differing computational and numerical techniques, the outcomes of PLS at both ends and IMS or DMS are unlikely to be exactly equal. Certainly in small samples one may find differences. Below we will therefore also report on the boundary cases IMS and DMS when the parameters are estimated using OLS. 4. Illustrations To illustrate the PLSAR model, consider the quarterly index of US industrial production, for the period January 1945 to April 2000 (the data source is Datastream). The data have been seasonally adjusted (using the familiar Census X-11 method), and transformed using the natural logarithmic transformation and after that using first differences. For the full sample, the estimated (partial) autocorrelation functions of these growth rates suggest an autoregressive order of 5, which in our notation means that the value of p is set at 5. Next, it is assumed that there is an interest in forecasting one to five quarters ahead. There is no constant in the models as all variables in the PLSAR model are scaled towards z-scores prior to analysis. 4.1 The Models To compare models on their forecasting performance we first follow a recursive procedure. First, the sample January 1945 to April 1990 is used, and forecasts are made for January 1991 to January Then the sample moves one quarter, i.e. it becomes January 1945 to January 1991 and again one- to five-step-ahead forecasts are made. This results in 40 one-step-ahead forecasts, 39 two-step-ahead forecasts and finally 36 five-step-ahead forecasts. Each time, the model parameters are reestimated. As said, before estimation, the variable is standardized to have mean zero and scaled by the standard deviation for the estimation sample. A second exercise that we do is that the estimation sample is kept fixed at 120 observations, i.e. a moving window sample. Each time the model moves on in time with one step, and the first observation is dropped. We use these two methods as these are commonly used in forecasting practice, and also as we want to examine the robustness of the forthcoming results. The forecasting experiment concerns a comparison of the same AR(5) model for all forecast horizons (IMS) with five versions of the PLSAR(5, 5) model. Each of these five versions assumes a different amount of latent variables. The case with five such latent variables is close to assuming five different AR(5) models for the five different forecast horizons (the DMS method), but not exactly equal. Hence we report on all findings. For the last estimation sample, the full PLSAR model gives latent explanatory variables which contribute the fractions 0.376, 0.349, 0.139, and of

8 396 FRANSES AND LEGERSTEE Table 1. Forecast Results for Quarterly Seasonally Adjusted Industrial Production in the USA. Horizon h IMS PLS 1 PLS 2 PLS 3 PLS 4 PLS 5 DMS Recursive samples Moving window samples Root mean squared prediction errors are given for an AR(5) model for all forecast horizons (IMS), PLSAR(5, 5) models with the number of latent variables being 1, 2, 3, 4 or 5, and finally, the AR(5) model with different parameters for different forecast horizons, denoted as DMS. Boldface numbers are the smallest in the row. the variance of X. Hence, based on these values one would tentatively set k equal to 2, at least for this sample. 4.2 Results The first set of forecasting results are displayed in Table 1. The first panel gives the results for the case of recursive samples, and the second panel deals with the 120-quarters moving window. With PLS k we denote a PLSAR with k latent variables for k = 1tok = 5. The last two columns are the case with k = 5andtheDMS method. The first column is the AR(5) model with the same parameters for each forecast horizon h (the IMS method), which comes close but is not exactly similar to PLS 1. To save space, the in-sample fit is not reported, but it is comparable with the numbers in the tables. This also holds for the other countries to be analysed below. For the recursive samples, the results are quite interesting. For horizons 2, 3, 4 and 5, the PLSAR models with k = 1 are better relative to both competitors. Note that this would not be the selected model based on the explained fractions of the variance though (of the last estimation sample). So, here we foresee an interesting further research topic. For the moving samples, the results suggest that most often the IMS method gives the most accurate forecasts. Different models for different horizons (DMS) are never the best. This seems to echo the findings in Kang (2003) who finds that models like equation (6) are not very successful in practice.

9 UNIFYING VIEW ON MULTI-STEP FORECASTING USING AUTOREGRESSION Empirical Results for More Countries The above exercise is repeated for similar seasonally adjusted industrial production series (quarterly) for Italy, Germany, Canada, Spain, Finland, France, Greece, Japan, Luxemburg, the Netherlands, Norway, Austria, Portugal, Sweden, Switzerland and the UK, which with the USA make a total of 17 series. A summary of an investigation of forecast accuracy is given in Table 2 for the recursive samples and moving window samples. Table 2 indicates that for the recursive samples the IMS method is best in 20.0% of the 85 cases, while the DMS is seldom preferred (3.5%). In fact, in 76.6% of the cases, a PLSAR model yields lowest root mean squared prediction errors. If we add the results of PLS 1 to those of IMS (56.5%), and we add the results of PLS 5 to those of DMS (9.4%), then a pure PLSAR model is best in 34.2% of the cases, which still seems to be a non-negligible fraction. Similar results hold for the moving window samples. A PLSAR model improves upon the IMS and DMS methods in 73.0% of the cases, and pure PLSAR models in 36.5% of the cases. These results add to the finding that DMS is not often improving on IMS. Also, the IMS approach can be improved by including one or a few more latent factors. 4.4 Further Empirical Results Until now we used five AR lags and five horizons. To see if the results obtained so far also hold for other AR orders, we examine the cases where p is set equal to 4 Table 2. Forecast Results for the IMS, PLSAR(5, 5) and DMS Methods for 17 Quarterly Seasonally Adjusted Industrial Production Series. Horizon h IMS PLS 1 PLS 2 PLS 3 PLS 4 PLS 5 DMS Recursive samples Fraction (of 85 cases) Moving window samples Fraction (of 85 cases) The cells contain the number of cases (out of 17) where the root mean squared prediction errors are smallest. The value of p is 5.

10 398 FRANSES AND LEGERSTEE Table 3. Forecast Results for the IMS, PLSAR(5, 4) and DMS Methods for 17 Quarterly Seasonally Adjusted Industrial Production Series. Horizon h IMS PLS 1 PLS 2 PLS 3 PLS 4 DMS Recursive samples Fraction (of 85 cases) Moving window samples Fraction (of 85 cases) The cells contain the number of cases (out of 17) where the root mean squared prediction errors are smallest. The value of p is 4. Table 4. Forecast Results for the IMS, PLSAR(5, 6) and DMS Methods for 17 Quarterly Seasonally Adjusted Industrial Production Series. Horizon h IMS PLS 1 PLS 2 PLS 3 PLS 4 PLS 5 PLS 6 DMS Recursive samples Fraction (of 85 cases) Moving window samples Fraction (of 85 cases) The cells contain the number of cases (out of 17) where the root mean squared prediction errors are smallest. The value of p is 6.

11 UNIFYING VIEW ON MULTI-STEP FORECASTING USING AUTOREGRESSION 399 Table 5. Forecast Results for the IMS, PLSAR(10, 5) and DMS Methods for 17 Quarterly Seasonally Adjusted Industrial Production Series. Horizon h IMS PLS 1 PLS 2 PLS 3 PLS 4 PLS 5 DMS Recursive samples Fraction (of 85 cases) Moving window samples Fraction (of 85 cases) The cells contain the number of cases (out of 17) where the root mean squared prediction errors are smallest. The case of the AR(5) model and horizon 10. and 6. The results in Tables 3 and 4 suggest that the results are fairly robust across the choice for p. This does not suggest that the model order is not important, but merely that the general finding that IMS and DMS can be improved by PLSAR models is robust. The same holds when we compare PLSAR(10, 5) models, i.e. when we consider forecast horizons beyond the autoregressive order (see Table 5). Again, similar fractions of success can be reported, although it now becomes apparent that for further away horizons the IMS method dominates. 5. Conclusion This paper has put forward a simple autoregressive time series model that can be used to jointly predict 1 to h steps ahead. The model has two specific cases as

12 400 FRANSES AND LEGERSTEE boundary cases, one is an AR model with the same parameters for all horizons and the other is an AR model with different parameters for different forecast horizons. The illustrative results showed that the resultant PLSAR can deliver more accurate forecasts than these two specific boundary cases. Hence, this new unifying model deserves further analysis and application. There are quite a number of further issues that need to be studied, additional to extensive applications to other time series. The first is that this paper only looked at point forecasts and not at forecast densities. Perhaps bootstrapping-based techniques can be used to retrieve this forecast distribution based on the errors defined by Ŷ X ˆB pls. Also, PLS in other social sciences is typically used and compared to other methods such as PCA and Lisrel. It is of interest to see how these methods would perform in a time series context. A next important research area concerns the development of an evaluation criterion that jointly evaluates forecast quality at all horizons, in order to compare the different model orders. We assumed that h and p are given, but a criterion that properly selects the value of p is of interest. Clements and Hendry (1993) address a related issue that is a model selection criterion which is specifically designed to jointly compare models for levels and growth rates, and perhaps a PLSAR order selection criterion could be like that one. Another interesting area concerns the derivation of a formal test for the rank of ˆB pls, which would allow for a choice between the various PLSAR models. Finally, this paper dealt with a single time series, but one can easily extend the PLSAR model to the case of two or more variables. Even more interesting would be to put forward a method to analyse one or more such series when they are nonstationary and have a unit root. At present, the most popular test for cointegration is based on CCA (see Johansen, 1995), and perhaps a PLS-based method would give more power to the test procedure. Like OLS, PLS can always be used, but it is likely that the statistical properties of the estimators are different from those in the case of stationary time series. Acknowledgements The encouraging comments made by the editor (Professor Oxley) and the detailed comments made by the two anonymous referees are very much appreciated, as well as the help of Dick van Dijk with collecting the data. The computed code that was used for all computations in this paper can be obtained from the authors. Notes 1. Further readings on PLS and related techniques include Helland (1990) and Wold (1966). The last author is said to be the first to have put forward the PLS technique. PLS has been used in other situations, notably for combining regressors when there are many see for example Garthwaite (1994), Naik and Tsai (2000) and Stone and Brooks (1990). 2. It is assumed that there are n effective observations, and hence that the full sample contains n + p + h 1 observations where the first p are needed for start-up.

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