Sanjib Bordoloi, Dipankar Biswas, Sanjay Singh
Ujjwal K. Manna and Seema Saggar*

During the recent period, dynamic factor modelling is gaining importance as one of the key forecasting tools exploiting the information contained in large datasets. The major advantage with the factor modelling approach is that, it can cope with many variables without running into scarce degrees of freedom that often arise in regression analysis. This technique allows forecasters to summarize the information contained in large datasets and extract a few common factors from them. This study attempts to develop a dynamic factor model (DFM) to forecast industrial production and price level in India. For this purpose, domestic as well as external economic indicators, that appear to contain information about the movement of industrial production/ price level, were used. Based on empirical analysis, it is found that the out-of-sample forecast accuracy of DFM, as measured by root mean square percentage error, is better than the OLS regression.

JEL Classification : C3, E2, E3.

Keywords : Dynamic Factor Model, Forecast, inflation, Output

Introduction

Reliable forecast of key macro economic indicators plays an important role for the formulation of monetary and fiscal policies of a country. Academic work on macroeconomic modeling and economic forecasting historically has focused on models with only a handful of variables. In contrast, economists in business and government, whose job is to track the swings of the economy and to make forecasts that inform decision-makers in real time, have long examined a large number of variables. Practitioners use many series when making their forecasts, and despite the lack of academic guidance about how to proceed, they suggest that these series have information content beyond that contained in the major macroeconomic aggregates. But if so, what are the best ways to extract this information and to use it for real-time forecasting? In the same line, Stock and Watson (2006) had focused on the case of United States, where, literally thousands of potentially relevant time series were available on a monthly or quarterly basis.

During the recent period, one of the key forecasting tools, which is gaining importance for dealing with large datasets, is dynamic factor modelling. The major advantage with the factor modelling approach is that, it can cope with many variables without running into scarce degrees of freedom that often arise in regression analysis. Besides this, by using factor models, the idiosyncratic movements, which possibly include measurement error and local shocks, can be eliminated. Through factor analysis, one can extract the unobserved factors that are common to the economic variables and these can be used for real time dynamic forecasting. For instance, Stock and Watson (1989) used a single factor to model the co-movements of four main macroeconomic aggregates. DFM plays an important role in the theory of Capital Asset Pricing Model (CAPM) also, as asset returns are often modeled as a function of market risk, where market risk is a common factor explaining the returns of many assets. Similarly in Arbitrage Pricing Theory (APT), returns are considered as function of other indicators like market return, inflation risk, liquidity risk etc as well as of some idiosyncratic component. This gives a more reliable signal for policy makers and prevents them from reacting to idiosyncratic movements. The uses of dynamic factor models have been improved by recent advances in estimation techniques [Stock and Watson (2002); Forni, Hallin, Lippi and Reichlin (2005) and Kapetanios and Marcellino (2004)].

These techniques allow forecasters to summarize the information contained in large datasets and extract a few common factors from them. All or a subset of the estimated factors are then entered into simple regression models to forecast the economic indicators.

The co-movement of contemporaneous economic variables may be due to the fact that they are driven in part by common shocks. This allows parsimonious modeling while corresponding to the notion of binding macroeconomic comovement. To overcome the problem of degrees of freedom for estimation of an economic system, reduction of dimensions has gained importance in the recent period. Factor analysis allows for dimension reduction and has become a standard econometric tool for both measuring comovement and forecasting macroeconomic variables.

The primary objective of the study is to forecast the inflation and output growth using dynamic factor models. The complete list of indicators that have been considered for empirical analysis is provided in the Annexure. The indicators cover the various sectors of the economy, viz., monetary and banking, financial, price, real and external. Also the performance of the dynamic factor model has been compared with alternative methods like the time series and econometric techniques.

The remainder of the study is divided into three sections. Section I describes briefly the methodology of Dynamic Factor Model (DFM). The empirical results related to DFM and assessment of the forecast performance are discussed in Section II. Finally, Section III concludes.

Section I
Methodology of the Dynamic Factor Model

Factor models have a long history of use in cross-sectional settings, and their generalization to dynamic environments is due to Sargent and Christopher (1977), Geweke (1977) and Watson and Engle (1983). Important recent contributions include Stock and Watson (1989, 1991, 1993) and Quah and Sargent (1993), among others. The dynamic factor model of Stock and Watson (1991) was developed as a modern statistical framework for computing a composite index of coincident indicators.

Given a data set, one can divide it into a common part, which captures the comovements of the cross section and a variable specific idiosyncratic part. A vector of N variables is represented as the sum of two unobservable orthogonal components, viz. a common component, driven by few (fewer than N) common factors, and an idiosyncratic component, driven by N idiosyncratic factors. If we have only one common factor, affecting only contemporaneously all of the variables, such a factor can be interpreted as the reference cycle (Stock and Watson, 1989). These models imply that the economic activity is driven by some few latent-driving forces, which can be revealed by the estimation of the dynamic factors. However, it may be noted that the factors, their loadings, as well as the idiosyncratic errors are not observable.

Alternatively, in the frequency domain, the dynamics among a number of important economic variables can be characterized by high pairwise coherences at the lower business cycle frequencies. Dynamics in frequency domain can be observed through cross spectral density function, which presents the same dynamic information. The cross spectral density matrix decomposes variation and covariation among variables by frequency, permitting one to concentrate on the dynamics of interest (e.g. the business-cycle dynamics correspond to periods of roughly 2-8 years). Transformations of both the real and imaginary parts of the spectral density matrix have immediate interpretation in business-cycle analysis - the coherence statistic between any two economic variables effectively presents the strength of their relationship at different frequencies, while the phase statistic presents the lead/lag relationships at different frequencies.

However, a factor model must have two characteristics. First, it must be dynamic to capture the structural changes in the economy. Secondly, it must allow for cross-correlation among idiosyncratic components, since orthogonality is an unrealistic assumption for most applications. The model we propose to use in this project has both the characteristics. It encompasses as a special case of the static ‘approximate factor model’ of Chamberlain (1983) and Chamberlain and Rothschild (1983), which allows for correlated idiosyncratic components. It also generalizes the factor model of Sargent and Sims (1977) and Geweke (1977), which is dynamic in nature, but has orthogonal idiosyncratic components. An important feature of this model is that the common component is allowed to have an infinite Moving Average (MA) representation, so as to accommodate for both autoregressive (AR) and MA responses to common factors. In this respect, it is more general than a static factor model where lagged factors are introduced as additional static factors, since in such model AR responses are ruled out.

Analysis of co-movement in dynamic settings typically makes use of two nonparametric tools, viz., the autocorrelation function and the spectral density function. In the time domain, one examines multivariate dynamics through the autocorrelation function, which estimates the correlations of each variable with its own past as well as with the past of other economic variables in the system. As an example, one can characterize the dynamics of output, consumption, investment, net exports, money and prices across different countries over the years.

The parameters of the DFM can be estimated by maximum likelihood using the Kalman filter and the dynamic factors can be estimated using the Kalman smoother [Stock and Watson (1989, 1991)].

h-step ahead forecast:

Section II
Empirical Estimates

The study uses monthly data covering the period from April 1994 to March 2008 consisting 168 sample points. The list of variables with description and sources are provided in the Annexure. To test the forecasting performance of the alternative methods, the whole sample period is divided into two sub-samples, viz., in-sample and out-of-sample. The in-sample, covering the period from April 1994 to March 2007, is used to estimate the parameters, while the last twelve points from April 2007 to March 2008, were used to test for the out- of-sample forecasting performance.

2.1. Model for Industrial Production

2.1.1. Estimates of the model

For developing a dynamic factor model to forecast the monthly industrial production in India, thirteen economic indicators were selected. The estimates cover the sample from April 1994 to March 2007. Table-1 presents the list of selected indicators.

Based on these selected thirteen indicators, factor analysis has been performed and obtained thirteen factors. Table-2 presents the estimates of the initial eigen values along with the percentage of total variance explained corresponding to these eigen values. For determining the number of factors that to be retained for further analysis, we have applied the rule based on eigen values-greater-than- one. The factors with eigen values greater than 1.0 are considered signifi cant, explaining an important amount of the variability in the data, while eigen values less than 1.0 are considered too weak, not explaining a significant portion of the data variability. Based on this rule, the first six eigen values were selected, which together explained 62.7 percent of the total variation. The selected first six factors were than rotated through the application of Varimax method. The Component Score Coeffi cient Matrix is presented in Table-3.

2.1.2. Out-of-sample forecasting

As mentioned earlier, the last twelve data points covering the sample from April 2007 to Mach 2008, has been used to test for the out-of-sample forecasting performance of the model. A comparison has been made between the out-of-sample forecasting performances of the DFM with a simple equation based on the ordinary least square (OLS) regression of the estimated factors. Table-4 presents the forecast errors (measured as percentage of actual industrial production), based on the two alternative methods, along with the Root Mean Square Percent Error (RMSPE). The RMSPE of the DFM is found to be 2.45 percent, which is significantly lower than 3.87 percent based on the OLS regression, indicating better explanatory power of the DFM than the OLS method.

2.2. Model for Price Level / Inflation

2.2.1. Estimates of the model

For developing a dynamic factor model to forecast the monthly inflation in India, nine economic indicators were selected. Table-5 presents the list of selected indicators.

Based on these selected nine indicators, factor analysis has been performed and accordingly nine factors were extracted initially. Table-6 presents the estimates of the initial eigen values along with the percentage of total variance explained corresponding to these eigen values. For determining the number of factors to be retained for further analysis, eigen values-greater-than-one rule has been applied as done previously. Based on this rule, the first four eigen values were selected, which together explained 60.1 percent of the total variation. The selected first four factors were than rotated through the application of Varimax method. The Component Score Coeffi cient Matrix is presented in Table-7.

2.2.2. Out-of-sample forecasting

As done earlier, the out-of-sample forecasting performance of the DFM has been compared with that of a simple OLS regression based on the selected four factors. Table-8 presents the forecast errors (measured as percentage of actual WPI), based on the two alternative methods, along with the Root Mean Square Percent Error (RMSPE). The RMSPE of the DFM is found to be 1.14 percent which is marginally lower than that of the OLS regression based estimate. This supports the better explanatory power of the DFM than the OLS method.

Section III
Conclusion

This study explores to develop dynamic factor models (DFM) to forecast industrial production and price level in India. For this purpose, economic indicators that contain information about the future movement of industrial production/ price level are selected. These indicators chosen represent both domestic as well as external factors. Based on empirical analysis, it appears that the performances of DFM are quite encouraging. It is found that the out-of-sample forecast accuracy of DFM, as measured by root mean square percentage error, is better than the OLS regression.

^* Authors are working in the Department of Statistics and Information Management, Reserve Bank of India. The views expressed in the paper are those of authors and do not necessarily represent those of the RBI. Errors and Omission, if any, are the sole responsibility of the authors.

References

Chamberlain, G. (1983). “Funds, factors and diversification in Arbitrage Pricing Models”, Econometrica, 51,1305-1324.

Chamberlain, G., and M. Rothschild (1983). “Arbitrage factor structure, and mean variance analysis of large asset markets”, Econometrica, 51,1281-1304.

Forni M., Hallin M., Lippi M. and Reichlin L.(2005). ‘The generalized dynamic factor model: one-sided estimation and forecasting’, Journal of the American Statistical Association, 100, 830 – 840.

Geweke, J. (1977) “The Dynamic Factor Analysis of Economic Time-Series Models,” in D.J.Aigner and A.S. Goldberger (eds.), Latent Variables in Socioeconomic Models, (Amsterdam: North-Holland), 365-383.

Kapetanios G. and Marcellino M. (2004). ‘A parametric estimation method for dynamic factor models of large dimensions’, Queen Mary University of London Working Paper, No. 489.

Quah, D. and Sargent, T. (1993) “A Dynamic Index Model for Large Cross Sections”, Centre for Economic Performance Discussion Paper No. 132, London School of Economics.

Sargent, T.J., and Christopher S. (1977), “Business Cycle Modeling Without Pretending to Have Too Much a Priori Theory,” in C. Sims (ed.), New Methods of Business Cycle Research (Minneapolis: Federal Reserve Bank of Minneapolis, 1977).

Stock, J. H., and Watson, M. W., (1989) “New Indexes of Coincident and Leading Economic Indicators,” in O. Blanchard and S. Fischer (eds.), NBER Macroeconomics Annual (Cambridge, Mass.: MIT Press, 1989), 351-394.

--------------------------------- (1991) “A Probability Model of the Coincident Economic Indicators,” in K. Lahiri and G.H. Moore (eds.), Leading Economic Indicators: New Approaches and Forecasting Records (Cambridge: Cambridge University Press, 1991), 63-89.

----------------------------------- (1993), “A Procedure for Predicting Recessions with Leading Indicators: Econometric Issues and Recent Experience,” in J.H. Stock and M.W. Watson (eds.), Business Cycles, Indicators and Forecasting (Chicago: University of Chicago Press for NBER, 1993), 255-284.

----------------------------------- (1998), “Median unbiased estimation of coefficient variance in a time varying parameter model”, Journal of the American Statistical Association, 93,349-358.

----------------------------------- (1999), “Forecasting Inflation”, Journal of Monetary Economics, 44, 293-335.

----------------------------------- (2002), ‘Forecasting using principal components from a large number of predictors’, Journal of the American Statistical Association, 97, 1167 – 1179.

----------------------------------- (2006), ‘Forecasting with Many Predictors’, Handbook of Economic Forecasting, 1, 515-554

Watson, M. W., and Engle, R. F. (1983), “Alternative Algorithms for the Estimation of Dynamic Factor, Mimic and Varying Coefficient Models,” Journal of Econometrics, 15, 385-400

Annexure