การคาดการณ์ อัตราแลกเปลี่ยนด้ วยแบบจาลอง Dynamic Factor Model
Exchange Rate Determination with Dynamic Factor Model
สุมณี ศุภกรโกศัย1
Sumanee Suppakornkosai1
ABSTRACT
This paper revisited the seminal work of Meese and Rogoff (1983). They found that a random
walk model actually performs as well as any leading indicator models in out-of-sample exchange rate
forecasts. Using the dynamic factor model by Stock and Watson (2002), we could reassess exchange
rate prediction using a wider set of information that included variables from both macroeconomic sector
and financial market. Our large number of variables from Japan, Sweden, U.K. and U.S. could be
summarized into a small number of estimated factors by the principal components analysis, and then the
estimated factors together with the exchange rate were estimated by the Ordinary Least Squares. The
mean square error and mean absolute error were used to compare out-of-sample exchange rate
predictability. The results showed that the forecasting accuracy of the dynamic factor model significantly
improved on the random walk. Therefore, the dynamic factor model can possibly outperform the random
walk in other out-of-sample exchange rate determination exercises in the future.
Key Words: Dynamic Factor Model, Exchange Rate Forecasting, Model Evaluation
e-mail address: sumanee.s@bu.ac.th
Introduction
Since Meese and Rogoff (1983) remarked that the exchange rate models could not outperform a
simple random walk, it has been a major discussion for exchange rate determination studies although
three decades have passed. The random walk is still treated as a standard comparator to any attempt on
exchange rate prediction. A number of literatures re-accessed the seminal research, trying to come up
with new approaches that could beat the random walk.2 The forecasting success of economic models for
exchange rate determination typically required regular updates on variables, model specification and
คณะเศรษฐศาสตร์ มหาวิทยาลัยกรุงเทพ 119 ถนนพระราม 4 เขตคลองเตย กทม. 10110
School of Economics, Bangkok University, 119 Rama 4 Road, Klong-Toey, Bangkok 10110
2
For example, see Cheung et al. (2005) and Clarida et al. (2003).
1
parameter estimates that had to be suitable for sample periods. However, the failure of economic models
remains largely the case.
Our motivation to revisit Meese and Rogoff’s infamous finding is a development of a dynamic
factor model by Stock and Watson (2002)3. They summarized the information from a large number of
predictors into a small number of estimated factors. Forecasting a macroeconomic time series variable by
using the factor model, they concluded that the factor model outperformed small vector autoregressions
and other leading indicator models. Prior to a development of the factor model, a VAR technique had
been widely used in economic forecasting due to its simplicity. Meese and Rogoff (1983) also used the
VAR as a representative multivariate time series model in their paper. One of the major deficiencies of the
standard VAR model is that it can only be low dimensional. A number of variables included in the VAR
model are very small in order to save degree-of-freedom. Also, the standard VAR requires representing
some theoretical construction by a specific variable. For example, a real economic activity that is
regularly used for predicting inflation can be represented by, e.g., unemployment rate, capacity utilization
or output gap. The question is which variable is the best measure of a real activity for this purpose. Under
the dynamic factor model, it is possible to pool information from all of the candidate predictors to form a
set of estimated common factors and use them for prediction instead. As a result, the factor model
appears to be a useful alternative forecasting method.
Therefore, in this paper, the dynamic factor model is used to represent the economic model of
exchange rate prediction against the random walk. The ultimate goal is to find out if the dynamic factor
model can improve on the random walk, or it is still an inadequate tool to use for predicting the exchange
rate.
Our explanatory variables included in the factor model are selected based on the flexible-price
monetary model and the sticky-price monetary model developed in 1970’s. Due to the criticism on
explanatory power of variables from traditional exchange rate models, and together with the poor
performance found by Meese and Rogoff (1983), we also cover variables from the financial sector. Hua
and Rey (2006) introduced an alternative exchange rate model that included only financial market
variables. The combined series of macroeconomic-based and financial-based variables are then used for
extracting the estimated factors. We conduct extensive examinations through different scenarios built on
3
Stock and Watson (2002) used the term “Diffusion Indexes” in their paper. However, it has been referred to as the dynamic factor
model afterward.
differing number of factors. The objective is to investigate the sensitivity of the outcomes to a different
setup.
The performances of each competing model are compared by their out-of-sample forecasting
accuracies. Forecasts for the dollar/pound, dollar/krona and dollar/yen exchange rates are generated at
one month horizon. A rolling regression is used to re-estimate the parameter vector for every forecast, so
parameters of each model can be estimated on the basis of the most up-to-date data available.
The methodology used for comparing the out-of-sample forecasting accuracy initially draws from
two statistic criterions; mean square error (MSE) and mean absolute error (MAE). The optimal model is
that the one with the smallest MSE and MAE. The mean square error is perhaps the most commonly used
statistics for forecasting comparison.
Overall, the results from our dynamic factor model prediction improve upon the random walk. As
extended investigation is conducted through various scenarios, the outcomes tend to vary across
exchange rate pairs. Nevertheless, the random walk is indeed possible to beat.
This paper is organized as follows. The second part discusses the methodologies used in this
chapter; the dynamic factor model framework and the out-of-sample forecasting comparisons. The third
part explains the explanatory variable selection and briefly presents details on data series used for the
estimation. We then turn to discuss statistical results in the fourth part, and final part is conclusion.
Methodology
In this section, we begin with the dynamic factor model layout. We later discuss a method used
for comparing the out-of-sample forecasting predictability.
Dynamic Factor Model
The idea of replacing a large number of information by a small number of estimated factors has a
long tradition. However, we choose to focus on the dynamic factor model developed by Stock and
Watson (2002). Let Yt be an (M × 1) vector of variables that need to be predicted, and let Xt be an (N × 1)
information vector that contains many time series variables. Further assume that Xt can be compressed
into a small K number of estimated factors Ft (K × 1), so Ft can summarize the information in Xt, and
N>>K. The joint dynamics of Ft and Yt in the dynamic factor model are given by the following equation:
𝐹
𝐹
[ 𝑡 ] = 𝛷(𝐿) [ 𝑡 ] + 𝑈𝑡 ,
𝑌𝑡
𝑌𝑡
(1)
where 𝛷(𝐿) = 𝛷1 𝐿 + ⋯ + 𝛷𝑝 𝐿𝑝 , a matrix of conformable lag polynomial of finite order p in the lag
operator L, is a ((K+M)x(K+M)) matrix of coefficients, and Ut is ((K+M)x1) vector of error term with mean
zero and covariance matrix Σ𝑈 . The relationship between the informational time series Xt and common
factors Ft is represented as follows:
Xt = ΛFt + et ,
(2)
where Λ is an (N × K) matrix of factor loadings, and et is an (N × 1) vector of error terms that have zero
mean and weakly correlated4.
We use the method of a two-step principal component approach for our estimation. The principal
component analysis is a multivariate technique for inspecting relationship among several quantitative
variables. Given our data series Xt with N variables, we can compute N principal components. Each
principal component is a linear combination of the stationary form of the original variables, and
coefficients equal to the eigenvectors of the correlation or covariance matrix. The eigenvectors are
usually taken with unit length. The principal components are sorted by descending order of the
eigenvalues, which are equal to the variances of the components. Consequently, the first principal
component has the largest variance of any unit-length linear combination of the original variables, while
the last principal component has the smallest variance of any linear combination of the original variables.
By applying the two-step principal component analysis to the relationship between Xt and Ft in
equation (2.2), it provides a nonparametric way of uncovering the common space spanned by the factors
of Xt. In the first step, the space spanned by the factors is estimated by using the first K principal
components of Xt; thus, we get estimated factors, 𝐹̂𝑡 . The second step involves estimation of the VAR that
includes 𝐹̂𝑡 and Yt in equation (1). In our case, Yt contains only the exchange rate. As for the number of
estimated factors including in our estimation, various numbers of estimated factors are tried. In our case,
we choose to include up to three estimated factors from the principal component estimation.5
Out-of-Sample Forecasts
As the random walk serving as our benchmark model, performances from the random walk and
the dynamic factor model are compared by their out-of-sample forecasting accuracies. We follow the
methodology as described in Meese and Rogoff (1983) to compare models out-of-sample. The out-of-
4
We use the principal component analysis in our estimation. In the principal component estimation, the residuals are generally
correlated with each other.
5
There is no formal rule of how many factors should be included. We determine our number of factors by simply looking at principal
component scores. The first three components already account for more than fifty percent of the variance, while the fourth
component and so forth do not contribute as much.
sample accuracy is measured by two statistics: mean absolute error (MAE) and mean square error
(MSE).6 The equations are defined as follows:
Mean absolute error = ∑𝑃−1
− 𝑌̂𝑡+1+𝑖 |/𝑃,
(3)
|𝑌
𝑖=0 𝑡+1+𝑖
Mean square error = ∑
𝑃−1
𝑖=0
2
(𝑌𝑡+1+𝑖 − 𝑌̂𝑡+1+𝑖 ) /𝑃,
(4)
where P is the total number of forecasts in the projection period for which the actual Y(t) is known, and
𝑌̂(t) is the forecasting value. Although we present both MAE and MSE results, MSE is our key criterion
used for comparing forecasters. The optimal forecasting model is that the one with the smallest MSE and
MAE.
Data
According to Meese and Rogoff, our selected macroeconomic-based variables are derived from
two well-known exchange rate models: the flexible–price and the sticky-price monetary models, while the
selection of financial-based variables for our study is derived from Hua and Rey (2006).
The bilateral data on preferred variables are collected across countries; the United States vs.
Japan, Sweden and the United Kingdom, to facilitate comparisons. More details on data sources are
available upon request. The data include 168 monthly observations from period 1998:01 to 2011:12. Even
though there is slight variation from country to country, the variables are selected from the following
categories:
1. The bilateral spot exchange rates for the dollar/yen, dollar/krona and dollar/ pound.
2. Industrial production indices and production of total manufacturing indices for the real
income variables.
3. The base money, M 1 and M 2 for the money supplies.
4. Three month Treasury bill rate, discount rate, and interbank rate for the short-term interest
rates, and long-term bond yields for the long-term interest rates.
5. Consumer price indices and producer price indices for the inflation rates.
6. Stock prices for the equity returns.
7. Various type of asset flows for the net capital flows.
6
Our measures are slightly different from Meese and Rogoff (1983). In their paper, three statistics; mean error (ME), mean absolute
error (MAE) and root mean square error (RMSE) are suggested. Mean error measures bias while root mean square error is simply
the square root of mean square error.
All series except exchange rates and interest rates are seasonally adjusted by X-12 arima
process. For the purpose of stationarity, the augmented Dickey-Fuller test is applied to all variables. If a
unit root is found, transformation is performed such that each variable has zero mean and unit standard
deviation.
Since there is no strict rule of how many factors should be included in the factor model, a
different number of factors are tried in order to examine any changes in our statistics. We compare the
results of the out-of-sample forecasting accuracy between the random walk and the factor model with
three different specifications; the exchange rate explained with one factor, with two factors and with three
factors.
Results
The random walk is a benchmark model in our study. We investigate the exchange rate
determination using the dynamic factor model, and the spot exchange rate for dollar price of foreign
currency is the only observable variable in our dynamic factor model. All other variables are used for
extracting principal components. The corrected Akaike information criterion is used to determine a lag
length for the dynamic factor model. The lag orders differ among exchange rate pairs, varying from zero
to two.
For all countries, the hold–out samples are set to 31 observations. Each bilateral exchange rate is
initially estimated using data up through the first forecasting period, June 2009, then the data for July
2009 are added to the sample, and the parameters are re-estimated using rolling regressions. New
forecasts are generated at one month horizon.
Tables1 presents the mean absolute error and mean square error at one month horizon for 31
hold-out-samples.
First, consider the dollar/pound exchange rate. The factor model outperforms the random walk
(0.0012 MSE, 0.0264 MAE). Model 2F improves 89 percent upon the random walk on MSE, while model
1F improves 68 percent upon the random walk on MAE.
We next consider the dollar/krona exchange rate. The factor model has lower statistics than the
random walk. Model 2F has the lowest MSE (0.0000021) and MAE (0.0010665), 87 and 65 percent less
than the random walk’s MSE (0.000016) and MAE (0.003015), respectively.
Lastly, consider the dollar/yen exchange rate. Although the statistics become very small, the
factor model has better results than the random walk (3.85E-08 MSE, 1.61E-04 MAE), and model 3F
performs the best (5.44E-09 MSE, 6.20E-05 MAE).
It should be noted that the results presented above do not answer the question whether one
particular model is significantly better than the other models according to our statistical outcomes.
However, given the robustness of the dynamic factor models’ superior performances over different
setups, the dynamic factor models are better than the random walk model for forecasting exchange rates
at one month forecast horizon.
Table 1. Exchange rate forecasting with one month horizon
Exchange Rate Pair Exchange Rate Model
MAE
MSE
Dollar/Pound
Dollar/Krona
Dollar/Yen
RW
0.026414
0.00120081
Model 1F
0.00836633
0.0001294
Model 2F
0.00852158
0.00012902
Model 3F
0.00857711
0.00013394
RW
0.00301484
1.56E-05
Model 1F
0.00108114
2.12E-06
Model 2F
0.0010665
2.06E-06
Model 3F
0.0010919
2.09E-06
RW
1.61E-04
3.85E-08
Model 1F
6.85E-05
6.82E-09
Model 2F
6.85E-05
6.89E-09
Model 3F
6.20E-05
5.44E-09
Conclusion
This paper draws on the dynamic factor model to compete against the random walk for
forecasting three bilateral exchange rates; the dollar/pound, dollar/krona and dollar/yen. The
methodology of the dynamic factor model is to summarize the information from a rich dataset into a small
number of estimated factors by the principal component approach, and then estimate the factors with the
exchange rate by the Ordinary Least Squares. Our macroeconomic-based variables are derived from the
structural models of the flexible-price and sticky-price monetary approaches, while financial-based
variables are subjected to Hua and Rey (2006). We compare one-month-ahead forecasting accuracy
based on rolling out-of-sample using mean absolute error and mean square error as our criterions. The
results of our study are uniformly strong; the factor models substantially improve on the random walk in all
exchange rate pairs. According to the outcomes, we conclude that the dynamic factor model can
possibly outperform the random walk in the out-of-sample exchange rate determination.
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