the data to have zero mean and unit variance. From this transformed and standardized data set we extract our macro-finance factors using PCA. The entire list of variables, details about their sources and their transformation are given in the appendix.10
Descriptive statistics of the macro factors are provided in Table 2.2.
Insert Table 2.2 about here
which implies that information of the comprehensive macroeconomic data set is compactly summarized in a strictly smaller number of factors.
We consider an approximate dynamic factor model as suggested in e.g. Stock and Wat-son (2002a) and Stock and WatWat-son (2002b) which allows to estimate the macro-finance factors conveniently by asymptotic PCA. Asymptotic PCA for data with a large cross section,N, and a small number of time series observations, T, were originally developed by Connor and Korajzcyk (1986). Approximate dynamic factor models are appealing due to their simplicity. The estimation of dynamic factor models is more complicated than the estimation its static counterpart.12 However, Boivin and Ng (2005) compare the forecast performance of the dynamic and static approach and conclude that both methods have similar forecast precision. Thus, we favor the static estimation via PCA in this paper.
We follow the approach suggested by e.g. Stock and Watson (2002a), Stock and Watson (2002b), Moench (2008) and Ludvigson and Ng (2010) to calculate principal components.
Accordingly, the factorsFt are defined by √
T times the r eigenvectors corresponding to ther largest eigenvalues of theT×T matrixy×y0. The factors are normalized such that Ft0Ft= Ir, whereIr is the identity matrix of dimension r and the eigenvalues are sorted in decreasing order. Intuitively, at each point in timet, the set of factor Ft is given by a linear combination of each element of theN ×1 vector yt = (y1t, . . . , yN t)0. The factors are chosen such that they minimize the sum of squared residuals of (yit−λ0iFt)2 as in a standard linear regression.
We denote the number of factors needed to summarize the information of the data set by r. In practice, the number of factors is unknown but Bai and Ng (2002) develop model selection criteria which are suited for a panel data setting. In particular, we rely on the below loss function to determine the appropriate number of factors:
IC =V(r, Fr) +rσ2
(N+T−r)ln(N T) N T
. (2.3)
The fit of a model with r + 1 factors cannot be worse than a model with r factors;
however, efficiency is lost as more factor loadings are estimated. In the above selection
12Even though the model specifies a static relationship betweenyit andFt,Ft may still be a dynamic vector process evolving according to A(L)Ft =ut where A(L) is a polynomial in the lag operator. We refer to Forni, Hallin, Lippi, and Reichlin (2005) for a detailed discussion of dynamic factor models.
criteria,V(r, Fr) is the sum of squared residuals from Equation 2.2 and measures the fit of the model whenr factors are estimated. The latter term of the loss function is a penalty which prevents us from overfitting and determines the number of factorsr. Note thatσ2 denotes a consistent estimate of (N T)−1PN
i=1
PT
t=1e2itand that the maximum number of estimated factors is set to 20.
The information criterion reported in Table 2.2 achieves its minimum ifr = 8. Thus, the data set is appropriately summarized by 8 factors, i.e. r = 1,2, . . . ,8. These 8 factors account for about 54.5% of the variance in the panel data set. The variance explained by each factor decreases in r since they are sorted in a descending order according to their eigenvalues.
2.3.2 Predictive Regressions
We study the predictability of currency portfolio returns and the business cycle depen-dence of currency risk premia using standard predictive regressions (as in Ludvigson and Ng (2009)). We are particularly interested in whether macro-finance factors provide infor-mation beyond forward discounts (or equivalently interest rate differentials vis-a-vis the U.S.), which are shown to be very powerful predictors of currency returns in Lustig, Rous-sanov, and Verdelhan (2010). This predictability is at the heart of the forward premium puzzle of Fama (1984a) and Hansen and Hodrick (1983). We regresshperiod log currency excess returns on lagged predictor variables, which include the AFD, denoted asF Dt, and a set of macro-finance factors. The predictive regression reads as follows
rxt+h =α+βf dF Dt+βk,x0 Fk,t+εt+h, (2.4) where βf d denotes the coefficient of the AFD and βr,x indicates the coefficient for the set of macro-finance factors Fk,t included in the regression which represents a subset of Fr,t. The AFD, that is F Dt = ft−st is an equally weighted average of the individual forward discounts of the basket of currencies vis-a-vis the USD that we consider. Lustig, Roussanov, and Verdelhan (2010) find this variable to be a more powerful predictor of FX market returns than the portfolio-specific forward discount, echoing previous results in bond markets by Cochrane and Piazzesi (2005).
Even though the extraction of macro-finance factors already represents a substantial re-duction of dimensionality and compactly summarizes the information from many economic series in a fewrfactors, it is necessary to gauge which of these factors are actually relevant in predicting currency returns. We follow Ludvigson and Ng (2009) in applying a model selection approach guided by the Bayesian information criterion (BIC). With 8 factors there are possibly 28−1 forecast models based on different combinations of factors in the forecast model. Faced by this model uncertainty, we evaluate all possible forecast mod-els and calculate the BIC for each model, which penalizes highly parametrized modmod-els.
Finally, we present estimation results for the 5 best models with the lowest BIC, which provide a summary of models with significant predictive power and parsimony.
To assess the information content of macro-finance factors for future currency excess re-turns, we consider both short-term forecast horizons of one month (h= 1) and longer-term horizons of one year (h = 12). The long-horizon regressions exhibit serial correlation in the error term due to overlapping observations, which is a well-known problem in these types of regressions. To account for this issue, we use two common remedies, HAC robust standard errors by Newey and West (1987) which are based on the optimal number of lags following Andrews (1991) as well as Hansen and Hodrick (1980) standard errors (HH) which are computed with h lags. Besides autocorrelation, another common econometric pitfall in predictive regression is a potential bias of coefficients in finite samples due to the persistence of the typically used predictors (see Stambaugh (1999)). We account for this well-known problem based on a parametric bootstrap procedure which provides valid in-ference in small samples. The bootstrap procedure largely follows Mark (1995) and Kilian (1999).13
13The bootstrap procedure imposes the null hypothesis of non-predictability and assumes an autore-gressive structure for the predictive variableszt= F Dt, Ft0
. The data generating process is assumed to be given byrxt+1=β0zt+u1tandzt+1=γzt+u2t. Based on this model we generate a sequence of pseudo observationsrx∗t andz∗t using the estimated coefficients ˆβ0 and ˆγ and by drawing with replacement from the estimated residualsu∗1tand u∗1t. With these pseudo observations at hands we estimate our predictive regression and obtain a distribution of the test statistic of our interest, i.e. θ∗ = (α∗, βf d∗ , βk∗0)0. This distribution is then used to calculate bootstrapped p-values.
2.3.3 Out-of-sample Forecasting Approach
While in-sample analysis is useful for uncovering predictive relations and assessing the evolution of risk premia, an interesting question is whether this information is also useful for predicting out-of-sample. To evaluate the out-of-sample performance of the macroeco-nomic and financial predictors, we adapt the forecast procedure of Bai (2009). The main idea here is to select continuously good predictors based on their pseudo out-of-sample predictive performance during an evaluation period over which the model performance is evaluated. This adaptive prediction procedure reflects the uncertainty faced by an in-vestor in real-time and allows her the updating of beliefs continuously by re-considering the prediction models.
The investor faces uncertainty about the best model to predict the currency market. Usu-ally, model selection criteria such as AIC, BIC or R2 are based on in-sample information only, whereas interest typically lies in out-of-sample forecast accuracy. Thus, models cho-sen based on these criteria may suffer from a lack of predictive power. To overcome this shortcoming we use the predictive least squares principle (PLS) as a model selection criterion.
The picture below briefly summarizes the adaptive forecast procedure.
-
-0 L-m L T
Estimation Evaluation Prediction
Graphical Illustration of the Out-of-sample Forecast Procedure
Suppose at time t = L, an investor faces uncertainty about the appropriate predictor model to support his investment decision. One way to reduce the uncertainty about the correct specification of the predictor model is to evaluate the out-of-sample during a model selection period which we refer to as the “evaluation window”.
At time t = L−m, an investor predicts currency returns based on the information set available attaccording to the predictive regression given by Equation2.4. The first forecast at time t = L−m+ 1 is denoted with the dashed line. For the first forecast we define the estimation window to have a length of 120 months, which represents our initialization period. After that, the estimation window is expanding. To select the most accurate model we define a model selection period, that is the “evaluation window”. It has length m=L−t, which we set to 12 months. During that period we calculate the PLS for each of the forecasting models defined by a specific combination of factors Fkt. We sum the PLS during the evaluation window, i.e. from L−m+ 1 (first forecast in the evaluation window) to L (last forecast in the evaluation window). In particular, for the evaluation period the sum of the PLS is given as:
P LS(zi) =
N
X
t=N−m+1
rxt+h− α+βf dF Dt+βk,x0 Fk,t2
.
At the end of the evaluation window the investors evaluates theP LS(zi) for all possible forecast models and selects the model with the lowest PLS. This model is then used to make a prediction atL+ 1, indicated by the second dashed line in the above figure. Thus, the procedure selects different forecast models based on out-of-sample information, i.e.
based on information which is available at timet.
The length of the evaluation window is somewhat arbitrary. A longer evaluation window has more statistical power while a shorter window is better at depicting the dynamic changes of the economic conditions. The shorter the evaluation window, the better it captures recent developments in the economy. Hence there is a conflict between the length and the quality of the model selection. To analyze the impact of the length of the evaluation window we choose the evaluation window to be 12, 24 and 36 months; however, there are no qualitative differences with respect to the length of the evaluation window. Thus, in Section 2.4.5 we report the out-of-sample forecasting results based on a evaluation window of 12 months.
2.3.4 Forecast Evaluation
To evaluate the forecast performance of models augmented by macro-finance factors we compare their forecasts with two benchmarks. First, a forecast where we instead of relying on a model selection procedure use all the eight factors to forecast currency returns. We label this forecast a kitchen sink forecast. Second, we also make use of the AFD to predict currency excess returns. We dynamically compare the predictive power of the macro-finance factor to the benchmark forecasts based on the difference in cumulative prediction errors. This evaluation is beneficial since we are able to evaluate the time series patterns in the forecast performance.
Based on these benchmarks we calculate Theil’s U to evaluate the statistical power of the of the factor based model. Theil’s U is given by the root mean square error (RMSE) of the forecast based on macro-finance factors relative to the RMSE of the benchmark model such that a value smaller than one indicates that the model beats the benchmark in terms of forecast accuracy. To assess statistical significance we calculate bootstrapped p-values. The bootstrap procedure is a model-based wild bootstrap imposing the null of non-predictability by macro-finance factors.14