forecasts of inflation and the output gap (Garratt et al., 2011), UK monetary aggre- gates (Garratt et al., 2009) and inflation forecasts by Bayesian model averaging (Groen et al., 2013). The present study follows these pioneer researchers and employs both Bayesian estimation and real-time data for the study of inflation.
Point and density forecasts from a wide variety of models are combined using both time-varying weights and equal weights strategies, as it remains unresolved whether or not equal weights can produce better forecasts (e.g., Stock and Watson, 2004; Clark and McCracken, 2009; Jore, Mitchell, and Vahey, 2010). Our forecasting results sug- gest that compared with the traditional specification of the unemployment rate Phillips curve, models that include other macroeconomic variables with proper error specifica- tions can outperform univariate modelsDMAand DMS in both point forecasts and density forecasts.
The remainder of the chapter proceeds as follows. Section 3.2 describes the spec- ifications of time-varying coefficients models and the component models for inflation forecast combination. Section 3.3 provides a brief introduction to the real-time data and presents a full-sample estimation using US inflation data. Section 3.4 discusses the forecasting results of the model combination for point and density inflation fore- casts, univariate models, dynamic model averaging, and dynamic modeling selection. In Section 3.5 we conclude.
3.2
Component Models
We consider three broad classes of time-varying coefficient models with different specifications of error terms: (i) models with constant variance (TVC); (ii) models with stochastic volatility (TVC-SV); and (iii) models with moving average stochastic volatility (TVC-SVMA). Within each class of model, we consider eight specifications, each with a different measure of economic activities for full-sample estimation. Finally, seven specifications are used for forecasting inflation. In addition, for each inflation predictor, various lag structures are considered, such as from one to four single lags, and up to four more lags.
By combining all lag structures with the seven predictors, this model averaging ap- proach can be conducted forTVC,TVC-SVand TVC-SVMA, respectively. In the following subsections, first, the specifications of TVC,TVC-SV and TVC-SVMA are described, and then the eight inflation predictor candidates are described.
3.2.1 Time-Varying Coefficient Models
3.2.1.1 Constant Variance
A generic TVC model can be described as a generalized Phillips curve with time- varying coefficients: yt+k=β1,t+Σnj=0β2+j,txt−j+εyt, ε y t ∼ N(0,σ2y), (3.2.1) βt=βt−1+εβt, εβt ∼ N(0,Q), (3.2.2) βt=(β1,t β2,t · · · βn,t )′ ,Q0 = σ02β 1 · · · 0 .. . . .. ... 0 · · · σ20βn ,Q= σβ2 1 · · · 0 .. . . .. ... 0 · · · σ2βn ,
where k is the forecasting horizon. For the full sample estimation in Section 3.3.2, we set k = 0. Note that the subscripts in the specification represent the order of the coefficients and time points (e.g., β1,t toβn,t), while the superscripts indicate the underlying relationship between variables (e.g.,y and εy). xt is a vector of covariates that may include lagged values of inflation or inflation predictors. The model above incorporates both a time-varying intercept and regression coefficients.
In Equation (3.2.2), the intercept and coefficients are assumed to follow indepen- dent random walks (e.g., Clark, 2011; Clark and Ravazzolo, 2015). By allowing the coefficients to evolve gradually over time, this specification accommodates a slowly changing relationship between inflation and the explanatory variables. Atkeson and Ohanian (2001) criticize the ability of the Phillips curve models to forecast inflation as compared with random walk forecasts. However, the Phillips curve models they adopt all have constant coefficients, and they do not perform as well as random walk naive forecasts in some historical periods. It can be expected that a time-varying Phillips curve performs better than one with constant coefficients.
The covariance matrices Q0 and Q of β0 and βt respectively are assumed to be diagonal matrices, whereQ0andβ0are initial values ofQandβt, respectively. It indi- cates thatβ1,t· · ·βn,t have individual independent white noise disturbances with zero means and variancesεβ1
t · · ·ε βn t , respectively. Both ε y t and ε β
§3.2 Component Models 45
3.2.1.2 Stochastic Volatility
Next, we extend (3.2.1)-(3.2.2) to allow for stochastic volatility (e.g., Groen, Paap, and Ravazzolo, 2013): yt+k=β1,t+Σnj=0β2+j,txt−j+εyt, ε y t ∼ N(0,eht), (3.2.3) ht=ht−1+εht, εht ∼ N(0,σh2), (3.2.4) βt=βt−1+εβt, εβt ∼ N(0,Q), (3.2.5) whereβtandεβt are the same as those inTVC, but the variance ofεyt is time-varying. The variance of εyt is controlled by log stochastic volatility ht, which changes over time. Equation (3.2.4) presents the evolution of the stochastic volatility parameter, which is an instantaneous volatility component of the model. The log-volatilities in Equation (3.2.4) are initialized byh1 ∼ N(0,σ02h)with variance given in advance, and
htfollows random walk innovation.
3.2.1.3 Moving Average Stochastic Volatility
We further extend the SV model (3.2.3)-(3.2.5) using the framework in Chan (2013). The TVC-SVMA model is given below:
yt+k=β1,t+Σnj=0β2+j,txt−j+εyt, (3.2.6) βt=βt−1+εβt, β1 ∼ N(0,Q0), εβt ∼ N(0,Q), (3.2.7)
εyt =ωt+ψ1ωt−1+· · ·+ψqωt−q, ωt∼ N(0,eht), (3.2.8)
ht=ht−1+εht, εht ∼ N(0,σ2h), (3.2.9) whereβt and εβt are the same as those inTVC. Equation (3.2.8) presents the moving average error feature of the specification. It can be rewritten as a polynomial of the lag operatorL:
εt =ψ(L)ut,
where ψ(L) = 1+ψ1L+· · ·+ψqLq. For identification, we assume that all roots of
ψ(L)are outside the unit circle. We assumeq =1 for simplicity.
3.2.2 Inflation Predictors
We consider a total of eight inflation predictors for each time-varying coefficient model (TVC, TVC-SV, and TVC-SVMA). As mentioned, we use real-time data.
However, data availability limits the number of variables used. Motivated by the predic- tive performance of inflation predictors in Groen et al. (2013), we select eight variables that include both real economy activities and a nominal variable (M2). The real-time inflation predictors are from the Real-Time Data Set for Macroeconomists (RTDSM) database at the Federal Reserve Bank of Philadelphia. The inflation predictors are:
1. Real unemployment rate (UR),
2. Real capacity utilization rate in manufacturing (CUR), 3. Housing starts (HSTS),
4. Real imports of goods and services (IMP), 5. M2 growth rate (M2),
6. Real durable consumption growth (RCON), 7. Real residential investment (RINV),
8. Real output growth (ROUT).
Among these eight predictors, there are seven real economy activity predictors, and one nominal predictor, M2.
3.2.3 Lag Structure
For each inflation predictor, we consider different forms of lag structure: coincident, and as a leading inflation predictor (one quarter ahead, half-year ahead, three quarters ahead, and one year ahead). We also include different numbers of lags (see Stock and Watson (1999)). Explicitly, the list of models is as follows:
yt+k=β1,t+β2,txt+εyt, (3.2.10) yt+k=β1,t+β2,txt−1+εyt, (3.2.11) yt+k=β1,t+β2,txt−2+εyt, (3.2.12) yt+k=β1,t+β2,txt−3+εyt, (3.2.13) yt+k=β1,t+β2,txt−4+εyt, (3.2.14) yt+k=β1,t+Σ1j=0β2+j,txt−j+εyt, (3.2.15) yt+k=β1,t+Σ2j=0β2+j,txt−j+εyt, (3.2.16) yt+k=β1,t+Σ3j=0β2+j,txt−j+εyt, (3.2.17) yt+k=β1,t+Σ4j=0β2+j,txt−j+εyt, (3.2.18)