Functional Coefficient Models for Nonstationary Time Series Data

Functional Coefficient Models for Nonstationary Time Series Data Zongwu Cai Department of Mathematics & Statistics and Department of Economics, University of North Carolina at Charlotte, USA Wang Yanan Institute for Studies in Economics, Xiamen University, China E-mail:zcai@uncc.edu

Contents Review and Motivations Parametric and Nonparametric Models Econometric (Statistical) Modeling and Theory A Simple Empirical Example Discussions 1

Review and Motivation A nonlinear time series model for a forecasting can be expressed Y t = g(x t ) + ε t, (1) where Y t is the forecasting variable(s) and X t is a vector of predictor variables. In the last three decades, various forms of model (1) has been explored. See the books by Tong (1990), Granger and Teräsvirta (1993), and Fan and Yao (2003). In the literature, most of studies can be classified into nonlinear parametric form and purely nonparametric approach. Also, it is commonly assumed that all variables (particularly X t ) in model (1) are stationary, denoted by I(0). 2

Review and Motivation Model (1) is very useful in applications, particularly in economics and finance. For example, it can be used for forecasting the inflation rate in macroeconomics, and testing the predictability and stability of stock returns in finance. Although model (1) is useful, it might not have a good predictive power when X t is I(0). To illustrate the above phenomena, let me show you an example. The inflation rate is defined as the log return Y t = ln(p t ) ln(p t 1 ), where P t is the consumer price index. If P t is the stock price, then Y t is the stock log return. 3

Review and Motivations In both applications of forecasting the inflation rate and testing predictability of stock returns, the main purpose is to build an econometric (statistical) model to forecast Y t based on some information, X t, including the lagged variables and some exogenous/endogenous economic and financial variables. Svensson and Woodford (2003) summarized the empirical literature for forecasting the inflation rate and concluded that Under normal circumstances, the information content of money growth for inflation forecasts in the short and medium term seems to be low. Only in the long run does a high correlation between money growth and inflation result. 4

Review and Motivations In finance, numerous studies in the last two decades have been devoted to answering the question: whether stock returns can be predicted by financial variables the dividend-price ratio, the earnings-price ratio, various measures of the interest rate. See the papers by Torous, Valkanov and Yan (2004) and Campbell and Yogo (2006) and others. 5

Review and Motivations For example, for forecasting the inflation rate, most of the papers focus on linear vector autoregressive (VAR) models: X t = α + Φ(L) X t 1 + e t, where X t = (Y t, m t ), Y t is the inflation rate at time t, m t = m t m t 1 is the growth rate of a monetary aggregate m t, and Φ(L) is a lag polynomial of certain order. 6

Review and Motivations As pointed out by Bernanke, Boivin and Eliasz (2005), using VAR models in these empirical studies might lead to at least three potential problems. First, to the extent that central banks and the private sector have information not reflected in the VAR, the measurement of policy innovations is likely to be contaminated. Second, the choice of a specific data series to represent a general economic concept such as real activity is often arbitrary to some degree. Third, impulse responses can be observed only for the included variables, which generally constitute only a small subset of the variables that the researcher and policy-maker care about. 7

Review and Motivations Recently, Bernanke, Boivin and Eliasz (2005) proposed a factor-augmented VAR (FAVAR) model to properly identify the monetary transmission mechanism. The joint dynamics of x t and f t are given by the following transition equation: ( xt f t ) = Φ(L) ( xt 1 f t 1 ) + v t, where f t is a vector of unobserved factors, summary of additional information. 8

Review and Motivations The use of VAR modeling is due to the facts: it is an atheoretical approach and it requires very few assumptions. However, it has become widely accepted that, for most purposes, changes in monetary aggregates are of little interest for the monetary policy process; see Leeper and Roush (2003). Numerous evidences have shown that money growth has no or little predictive power for inflation, and this finding is robust to changes in the sample period and econometric methodology. See Svensson and Woodford (2003). 9

Review and Motivations As advocated by Campbell and Yogo (2006), a linear or nonlinear regression of inflation rate (stock return) onto lagged variables and some stationary variables (market returns) has a low predictive power because stationary predictors like returns are extremely noisy and/or highly persistent. Question: How to make an improvement on modeling and forecasting? 10

Review and Motivations As pointed out by Campbell and Yogo (2006), if some of this noise can be eliminated, the predictive power might be improved significantly. One of solutions is to use some less noisy variables such as integrated or nearly integrated variables and to use a model which has an ability of capturing the persistence. Question: What kind of less noisy variables should be used? What kind of model should be used to capture the persistence? 11

Review and Motivations For an AR(1) model x t = ρ x t 1 + u t, if ρ = 1, it is called unit root or integrated process I(1); if ρ = 1 + c/n, it is called the nearly integrated or local-to-unity process. Applications in Macroeconomics: Motivated by the P inflation forecasting model, we can evaluate the information content of velocity, which has been studied by many authors; see Gerlach and Svensson (2003). The velocity is calculated as V t = P t Q t /M t, where P t is the price level (consumer price index), Q t is the index of industrial production and M t is one of the six monetary aggregates; see Cochrane (1994). 12

Review and Motivations Econometric (statistical) issues: It is well known (by the augmented Dickey-Fuller test) that V t is nonstationary, e.g. unit root I(1). But, the use of velocity to forecast inflation is lack of appropriate theory, i.e., no econometric theories can explain why a nonstationary covariate can help to forecast a stationary response variable (without any possible co-integration relationships) in a linear regression model framework, while in the literature, any econometric theory of applying nonparametric techniques to nonstationary and nonlinear data is not well developed yet. 13

Nonparametric Methods Without any appropriate econometric theory, Bachmeier, Leelahanon and Li (2007) considered the following general nonparametric model Y t = g(y t 1, X t ) + ε t, (2) where Y t is the rate of inflation and X t is a vector of other economic variables that are believed to affect inflation. 14

Nonparametric Methods Findings in Bachmeier et al. (2007): If X t is taken to be the money growth rate, the nonparametric model can improve the forecasting slightly. By choosing X t as the lagged value of velocity, X t = V t 1, Bachmeier et al. (2007) found that the nonparametric model (2) leads to a remarked improvement in forecasting inflation. Unfortunately, they did not provide any theoretical justification of using a nonstationary variable in their nonparametric regression model because the relevant asymptotic theory is unavailable. 15

Parametric Models Applications in Finance: As a special case, a parametric form of model (1) was used by Campbell and Yogo (2006) to do efficient tests of stock return predictability, which is the so called predictive regression model, Y t = β 0 + β 1 X t 1 + ε t, X t = ρ X t 1 + u t, (3) where Y t is a stock return and X t 1 is the financial variables at t 1 which can be chosen to be 16

Parametric Models the log dividend-price ratio (d-p), the log earnings-price ratio (e-p), the three-month T-bill, and the long-short yield spread. Paye and Timmermann (2006) considered the following model Y t = β 0,t + β 1,t X t 1 + ε t, X t = ρ X t 1 + u t, and instability test to test whether the coefficients are constant or not. 17

Parametric Models The main difficulties in a predictive regression model are: 1. Nonstationarity and Persistency: ρ = 1 + c/n means that X t is nonstationary [either nearly integrated or integrated] and persistent (see real examples and figures later). 2. Endogeneity: corr(ε t, u t ) 0 so that corr(x t 1, ε t ) 0 and X t 1 is an endogenous variable; see Table 4 of Campbell and Yogo (2006) and Table 1 in Paye and Timmermann (2006) for real examples in finance. 3. Nonlinearity between Y t and X t 1 (see Figure 7 later). 4. Instability: time-varying coefficients (see Figure 8 later). 18

ARTICLE IN PRESS J.Y. Campbell, M. Yogo / Journal of Financial Economics 81 (2006) 27 60 47 Table 4 Estimates of the model parameters Series Obs. Variable p d DF-GLS 95% CI: r 95% CI: c Panel A: S&P 1880 2002, CRSP 1926 2002 S&P 500 123 d p 3 0.845 0.855 ½0:949; 1:033Š ½ 6:107; 4:020Š e p 1 0.962 2.888 ½0:768; 0:965Š ½ 28:262; 4:232Š Annual 77 d p 1 0.721 1.033 ½0:903; 1:050Š ½ 7:343; 3:781Š e p 1 0.957 2.229 ½0:748; 1:000Š ½ 19:132; 0:027Š Quarterly 305 d p 1 0.942 1.696 ½0:957; 1:007Š ½ 13:081; 2:218Š e p 1 0.986 2.191 ½0:939; 1:000Š ½ 18:670; 0:145Š Monthly 913 d p 2 0.950 1.657 ½0:986; 1:003Š ½ 12:683; 2:377Š e p 1 0.987 1.859 ½0:984; 1:002Š ½ 14:797; 1:711Š Panel B: S&P 1880 1994, CRSP 1926 1994 S&P 500 115 d p 3 0.835 2.002 ½0:854; 1:010Š ½ 16:391; 1:079Š e p 1 0.958 3.519 ½0:663; 0:914Š ½ 38:471; 9:789Š Annual 69 d p 1 0.693 2.081 ½0:745; 1:010Š ½ 17:341; 0:690Š e p 1 0.959 2.859 ½0:591; 0:940Š ½ 27:808; 4:074Š Quarterly 273 d p 1 0.941 2.635 ½0:910; 0:991Š ½ 24:579; 2:470Š e p 1 0.988 2.827 ½0:900; 0:986Š ½ 27:322; 3:844Š Monthly 817 d p 2 0.948 2.551 ½0:971; 0:998Š ½ 23:419; 1:914Š e p 2 0.983 2.600 ½0:970; 0:997Š ½ 24:105; 2:240Š Panel C: CRSP 1952 2002 Annual 51 d p 1 0.749 0.462 ½0:917; 1:087Š ½ 4:131; 4:339Š e p 1 0.955 1.522 ½0:773; 1:056Š ½ 11:354; 2:811Š r 3 1 0.006 1.762 ½0:725; 1:040Š ½ 13:756; 1:984Š y r 1 1 0.243 3.121 ½0:363; 0:878Š ½ 31:870; 6:100Š Quarterly 204 d p 1 0.977 0.392 ½0:981; 1:022Š ½ 3:844; 4:381Š e p 1 0.980 1.195 ½0:958; 1:017Š ½ 8:478; 3:539Š r 3 4 0.095 1.572 ½0:941; 1:013Š ½ 11:825; 2:669Š y r 1 2 0.100 2.765 ½0:869; 0:983Š ½ 26:375; 3:347Š Monthly 612 d p 1 0.967 0.275 ½0:994; 1:007Š ½ 3:365; 4:451Š e p 1 0.982 0.978 ½0:989; 1:006Š ½ 6:950; 3:857Š r 3 2 0.071 1.569 ½0:981; 1:004Š ½ 11:801; 2:676Š y r 1 1 0.066 4.368 ½0:911; 0:968Š ½ 54:471; 19:335Š This table reports estimates of the parameters for the predictive regression model. Returns are for the annual S&P 500 index and the annual, quarterly, and monthly CRSP value-weighted index. The predictor variables are the log 19

Parametric Models From Table 4 (the seventh column) of Campbell and Yogo (2006), it can be seen that the log of d-p and the log of e-p are either unit root or nearly integrated and highly persistent. From Table 1 (the last column) of Torous, Valkanov and Yan (2004), it concludes that the log of dividend yield (default spread, book-to-market, term spread, and short-term rate) is either unit root or nearly integrated and highly persistent. A nonlinear (parametric) form of model (1) was considered by Polk, Thompson and Vuolteenaho (2006) for equity-premium forecasts. 20

944 Journal of Business TABLE 1 95% Confidence Intervals for the Largest Autoregressive Root of the Stochastic Explanatory Variables Series Sample Period k ADF 95% Interval Dividend yield 1926:12 1994:12 5 3.30 (.960,.996) 1926:12 1951:12 1 2.84 (.915, 1.004) 1952:1 1994:12 1 2.65 (.956, 1.004) Default spread 1926:12 1994:12 2 2.49 (.976, 1.003) 1926:12 1951:12 3 0.90 (.984, 1.015) 1952:1 1994:12 2 2.50 (.963, 1.004) Book-to-market 1926:12 1994:08 6 2.35 (.977, 1.003) 1926:12 1951:12 6 1.60 (.967, 1.013) 1952:1 1994:08 6 1.24 (.986, 1.008) Term spread 1926:12 1994:12 6 3.57 (.955,.992) 1926:12 1951:12 6 3.11 (.943,.999) 1952:1 1994:12 2 1.83 (.957, 1.012) Short-term rate 1926:12 1994:12 8 1.85 (.984, 1.004) 1926:12 1951:12 1 1.90 (0.955, 1.012) 1952:1 1994:12 7 1.90 (.974, 1.007) Note. This table provides 95% confidence intervals for the largest autoregressive root U of stochastic explanatory variables typically used in predictive regressions. The explanatory variables used are Dividend yield, Default spread, Book to market, Term spread, and Short-term rate. Dividend yield is the log real dividend yield, constructed as in Fama and French (1988). Default spread is the log of the difference between monthly averaged annualized yields of bonds rated Baa and Aaa by Moody s. Bookto-market is the log of Pontiff and Schall s (1998) Dow Jones Industrial Average (DJIA) book-tomarket ratio. Term spread is the difference between annualized yields of Treasury bonds with maturity closest to 10 years at month end and 3-month Treasury bills. Short-term rate is the nominal 1-month Treasury bill rate. The augmented Dickey-Fuller statistic is denoted ADF, and we follow Ng and Perron (1995) in determining the maximum lag length k. 21

Question and Motivations The question is why we do need to consider a nonlinear (nonparametric) model. To answer this question, let me show you two real examples for forecasting the US inflation rate in Example 1 and for testing predictability of stock returns in Example 2. 22

Example 1 Example 1: The data used here are the monthly data from February 1959 to April 2002 downloadable from the Federal Reserve Bank of St. Louis, at the following web site: http://www.stls.frb.org/, including some financial data. 23

0.005 0.005 0.015 Time Series Plot of Inflation 0 100 200 300 400 500 0 5 10 15 20 25 0.0 0.2 0.4 0.6 PACF 0.005 0.005 0.015 0 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 ACF Inflation vs lag 1 0.005 0.005 0.015 24

Price Level Industry Output 50 100 150 40 60 80 100 140 0 100 200 300 400 500 P_t 0 100 200 300 400 500 Q_t 1000 3000 5000 Monetary Aggregate 0 100 200 300 400 500 M2 0 2000 4000 6000 8000 Monetary Aggregate 0 100 200 300 400 500 M3 25

3.5 4.0 4.5 5.0 Time Series Plot of Velocities 0 100 200 300 400 500 M2 2.8 3.0 3.2 3.4 3.6 3.8 4.0 0 100 200 300 400 500 M3 40 50 60 70 80 90 40 50 60 70 0 100 200 300 400 500 M2D 0 100 200 300 400 500 M3D 26

0.2 0.2 0.6 1.0 0.2 0.2 0.6 1.0 ACF of Velocities 0 5 10 15 20 25 M2 0 5 10 15 20 25 M2D 0.2 0.2 0.6 1.0 0.2 0.2 0.6 1.0 0 5 10 15 20 25 M3 0 5 10 15 20 25 M3D 27

0 5 10 15 20 25 M2 0.2 0.2 0.6 1.0 PACF of Velocities 0 5 10 15 20 25 M2D 0.2 0.2 0.6 1.0 0 5 10 15 20 25 M3 0 5 10 15 20 25 M3D 0.2 0.2 0.6 1.0 0.2 0.2 0.6 1.0 28

0.005 0.005 0.015 0.005 0.005 0.015 Inflation Rate vs Velocity 3.5 4.0 4.5 5.0 M2 40 50 60 70 80 90 M2D 2.8 3.0 3.2 3.4 3.6 3.8 4.0 M3 40 50 60 70 M3D 0.005 0.005 0.015 0.005 0.005 0.015 29

Example 2 Example 2: Consider the stock return (y t, S&P500 CRSP weighted value) with financial variables (x t, log dividend-price ratio and log earning-price ratio). The sample period is 1926:4-2002:4 at quarterly frequency. From Campbell and Yogo (2006), both valuation ratios are persistent and even nonstationary, especially toward the end of the sample period. The 95% confidence intervals for ρ are [0.957, 1.007] and [0.939, 1.000] for the log dividend-price ratio and the log earnings-price ratio, respectively; see Panel A in Table 4 of Campbell and Yogo (2006). 30

Example 2 Campbell and Yogo (2006) and others considered the testing problem H 0 : β 1 = 0 by using the following model Y t = β 0 + β 1 X t 1 + ε t, X t = ρ X t 1 + u t, where Y t is a stock return, X t 1 is the financial variables at t 1 and ρ = 1 + c/n. To remove the endogeneity, using a projection ε t onto u t, Amihud and Hurvich (2004) considered the following model Y t = β 0 + β 1 X t 1 + γ u t + v t, X t = ρ X t 1 + u t, where ρ < 1. 31

Example 2 To capture the instability, Paye and Timmermann (2006) considered the following model Y t = β 0,t + β 1,t X t 1 + ε t, X t = ρ X t 1 + u t, where ρ < 1, and the testing problem for breaks as H 0 : β 1 = β 10 I(t T 0 ) + β 11 I(t > T 0 ). Are the above three models appropriate for the real applications? 32

Return 3.5 3.0 2.5 2.0 Return 4.5 4.0 3.5 3.0 2.5 Return vs log of d/p 0.4 0.2 0.0 0.2 0.4 log d/p Return vs log e/p 0.4 0.2 0.0 0.2 0.4 log e/p Return 3.5 3.0 2.5 2.0 Return 4.5 4.0 3.5 3.0 2.5 Return v.s. lagged log d/p 0.4 0.2 0.0 0.2 0.4 log d/p Return vs lagged log e/p 0.4 0.2 0.0 0.2 0.4 log e/p 33

0.05 0.10 0.15 0.20 0.25 0.30 Rolling Estimate of Slop beta log dividen price ratio 0 10 20 30 40 50 60 70 0.90 0.80 0.70 Rolling Estimate of Slop gamma log dividen price ratio 0 10 20 30 40 50 60 70 0.05 0.15 0.25 0.35 Rolling Estimate of Slop beta log earning price ratio 0 10 20 30 40 50 60 70 0.98 0.94 0.90 Rolling Estimate of Slop gamma log earning price ratio 0 10 20 30 40 50 60 70 34

Nonparametric Methods Model (2) can be regarded as a special case of the general nonparametric model (1). Also, model (3) can be re-written as Y t = β 0 + β 1 X t 1 + ε t = g(x t 1 ) + v t, where g(x t 1 ) = β 0 + β 1 X t 1 + E[ε t X t 1 ], v t = ε t E[ε t X t 1 ], E[v t X t 1 ] = 0, and X t 1 is either I(1) or nearly I(1). So, model (3) is a as a special case of model (1). 35

Nonparametric Methods There are many nonlinear forms for g( ) to be explored. We study the following general varying coefficient model Y t = p j=1 β j (Z t ) X jt + ε t, (4) where Z t is a vector of stationary or nonstationary variables or time t and X t is a vector of stationary or nonstationary variables. Model (4) can be regarded as an approximation of (1). 36

Nonparametric Models Model (4) covers several scenarios: (I) X t is I(1) and Z t is stationary, (II) X t is stationary and Z t is I(1); (III) Z t = t is a time trend variable and X t is I(1); see Park and Hahn (1999) and Chang and Martinez-Chombo (2003); (IV) both X t and Z t are I(1); (V) partially linear models; (VI) co-integration issues such as Y t and X t are I(1) but ε t is I(0). 37

Nonparametric Models Particularly, model (4) includes the threshold autoregressive (TAR) model studied by Bachmeier et al. (2007) Y t = p j=1 β j (V t 1, θ) Y t j + ε t, (5) where β j (v, θ) is a threshold function. Bachmeier et al. (2007) found that model (5) has more predictive power than model (2). 38

Nonparametric Models Also, model (4) can be regarded as a generalization/special case of the time-varying parameter (TVP) VAR model studied by Boivin (2001) for the stochastic coefficients, which is designed to capture the persistence. Finally, model (4) can be generalized to cover the polynomial terms of the integrated variable. 39

Nonparametric Models For simplicity, I consider the following simple varying coefficient model in scenario I: Y t = } β 1 (Z {{ t ) X t1 } + } β 2 (Z {{ t ) X t2 } +ε t stationary nonstationary = X T t β(z t ) + ε t, 1 t n, (6) where X t1, Z t, and ε t are stationary, X t2 is nonstationary such as an I(1) process, β(z t ) = (β 1 (Z t ), β 2 (Z t )) T, and X t = (X t1, X t2 ) T. 40

Nonparametric Models We apply the local linear fitting scheme to estimate the coefficient functions {β j ( )}. That is, for Z t in a neighborhood of the grid point z, β(z t ) β 0 + β 1 (Z t z). Then, the locally weighted least squares function is n [ Yt X T t β 0 X T t β 1 (Z t z) ] 2 K((Zt z)/h), (7) t=1 where K( ) is a kernel function and h is the bandwidth. 41

Asymptotic Theory Theorem 1: Under some regularity conditions, nhhn [ ˆβ(z) β(z) 1 ] 2 h2 µ 2 (K)β (z) d MN(Σ β (z)), where H n = diag{1, n} and MN(Σ β ) is a mixed normal distribution with mean of zero and some conditional covariance Σ β involving integrations of a standard Brownian motion. 42

Consequences of Theorem 1: Asymptotic Theory If there is no X t2, e.g. there is no nonstationary covariate, the results are the same as those in Cai, Fan and Yao (2000). The asymptotic bias for ˆβ j (z) is h 2 µ 2 (K)β j (z)/2, same as that for stationary case. Convergence rate for ˆβ2 (z) (coefficient function for nonstationary covariate) is faster than that of ˆβ 1 (z) (coefficient function for stationary covariate) by a factor of n 1/2. 43

Asymptotic Theory The asymptotic mean squared error (AMSE) for each estimator can be derived for ˆβ 1 (z), AMSE 1 = h4 4 µ2 2(K)[β 1(z)] 2 + σ β,11(z) n h where σ β,11 (z) is the first diagonal element of Σ β (z), and for ˆβ 2 (z), AMSE 2 = h4 4 µ2 2(K)[β 2(z)] 2 + σ β,22(z) n 2 h. 44

Asymptotic Theory By minimizing the AMSE with respect to h, we obtain the optimal bandwidth for ˆβ 1 (z), which h 1,opt = O(n 1/5 ) and the optimal AMSE, which is O(n 4/5 ). However, the optimal bandwidth for ˆβ 2 (z) is h 2,opt = O(n 2/5 ) and the optimal AMSE is O(n 8/5 ). 45

Asymptotic Theory This discussion implies that the optimal bandwidth of estimating β 1 (z) for is h 1,opt = O(n 1/5 ) and the optimal bandwidth of estimating β 2 (z) is h 2,opt = O(n 2/5 ). Clearly, if either one is used, the other one is not optimal. How to achieve the optimality? 46

Asymptotic Theory To estimate coefficient functions optimally, we suggest a twostage estimation procedure. The idea is similar to the profile least squares method; see Fan and Zhang (1999), Cai (2002) and Fan and Huang (2005). The first step is to estimate β 3 (z) using the profile least squares method and the second step is to estimate β 1 (z) and β 2 (z) using the pseudo residual. At both steps, a kernel smoothing technique is used. 47

Asymptotic Theory Theorem 2: Under some regularity conditions, [ n2 h 1 ˆβ 2 (z) β 2 (z) 1 ] 2 h2 1µ 2 (K)β 2(z) d MN ( σβ 2 2 (z) ), where MN(σ 2 β 2 (z)) is a mixed normal distribution with mean of zero and conditional variance σ 2 β 2 (z). And, h2 n [ ˆβ 1 (z) β 1 (z) 1 ] 2 h2 2µ 2 (K)β 1(z) d MN ( σβ 2 1 (z) ) for some σ 2 β 1 (z). 48

Asymptotic Theory This two-step method is oracle in the sense that the asymptotic result is same as that for the stationary case; see Cai, Fan and Yao (2000), or same as the case where β 2 ( ) were known. What are the asymptotic results for scenario II: X t is stationary and Z t is I(1)? For this case, the local linear fitting scheme is still applicable here. But the asymptotic result is totally different and the theoretical proof is very involved. 49

Theorem 3: Under some regularity conditions for scenario II, [ n 1/2 h ˆβ(z) β(z) 1 ] 2 h2 µ 2 (K)β (z) d MN(Σ β ), where MN(Σ β ) is a mixed normal distribution with mean zero and conditional covariance Σ β associated with the local time of a standard Brownian motion. Also, we considered the asymptotic behavior at boundary. 50

Asymptotic Theory Main Findings: 1. Asymptotic bias is the same as that for stationary case. 2. The rate of convergence is totally different with an extra factor n 1/4, which is slower than that for stationary case. 3. The optimal bandwidth is h opt = O p (n 1/10 ). 4. Asymptotic behavior at boundary is different from that for stationary case. 51

An Empirical Example We report an empirical application of using the varying coefficient model to forecasting the US inflation rate. We consider 1, 6, 12 and 24 months ahead forecasting of inflation. The data used are the monthly data from February 1959 to April 2002. The linear inflation forecasting model is Y t = Xt T β + ε t, (8) where Y t is the rate of inflation and X t contains lagged value of Y t and other stationary variables such as lagged value of money growth rate (say, m t 1 ). 52

An Empirical Example We consider the following varying coefficient model Y t = β 0 (Z t ) + β 1 (Z t )Y t s + β 2 (Z t )Y t s 1 + ε t, (9) where Z t = V t s, s = 1, 6, 12, 24. V t = P t Q t /M t, where P t is the price level (consumer price index), Q t is the index of industrial production, and M t is one of the six monetary aggregates. Augmented Dickey-Fuller unit root tests confirm that all the velocities are nonstationary. Also, it can be evidenced from Figure 4. 53

An Empirical Example We forecast inflation for the last 100 observations in the sample (January, 1994 to April, 2002). The mean square prediction error (MSPE) is defined as 2002:4 t=1994:1 [Ŷt Y t ] 2 /100, where Ŷt is the predicted value of Y t. We report relative MSPE obtained using model (9) to that based on the linear model (8). 54

An Empirical Example Here is the forecasting result. Table 1: Relative MSPE (Varying Coefficient Model/Linear Model) Forecast Horizon V2 V3 V2D V3D 1 Month 0.86 0.83 0.76 0.78 6 Months 0.92 0.94 0.86 0.88 12 Months 0.78 0.84 0.70 0.72 24 Months 0.89 0.72 0.72 0.71 55

An Empirical Example Table 1 shows that by adding a nonstationary covariate (velocity) and considering a flexible varying coefficient model structure, the inflation MSPE is reduced for all cases considered, and the reduction can be as much as 30%. This is in a sharp contrast to the cases of using only stationary covariates (say, adding a lagged money growth rate covariate) in forecasting inflation where a larger MSPE is usually obtained when compared to a simple linear forecasting model. 56

An Empirical Example The varying coefficient model also has an intuitive economic interpretation for the above model. Because the coefficients in (9) depend on the most recent observation on velocity, a varying coefficient model can be interpreted as one way of allowing for time-varying inflation persistence, which is important and has been the subject of numerous empirical papers; see Mankiw and Reis (2002). The degree of inflation persistence is determined by the monetary policy regime. (9) suggests that the policy regime is identified by velocity. 57

Discussions and Future Research We are working on the financial applications and hopefully, the results will come out very soon. Of course, I can send them to you through e-mail if you have an interest... 58

Discussions and Future Research Econometric (statistical) Issues: 1. How to select the bandwidths in this context? 2. How about the asymptotic theory for other aforementioned scenarios? 3. How about the testing problems related to the methodologies developed here? 4. How about the econometric results for other types of nonstationary variables such as nearly integrated? 59

Discussions and Future Research Economic (statistical) Issues: 5. If covariates contain some lagged variables, the stationarity of Y t must be addressed. 6. Empirical studies on comparisons with other models such as the Phillips curve model and the new Keynesian Phillips curve model. 7. Economic interpretation of using velocity in forecasting inflation rate based on a nonlinear model. 60

Discussions and Future Research Financial Applications: 8. How to apply the methodologies developed here to analyze financial data? 9. For financial applications, it happens that some of regressors might be correlated with the measurement error, which means that those regressors are endogenous ; see Table 1 of Campbell and Yogo (2006) for examples in finance. It is well known in the literature that the endogeneity makes econometric inferences much more difficult. 61

End THANK YOU! 62