The standard practice in vector autoregression (VAR) modeling is to select the lag length and variables to be included using a one-step-ahead model. That mo- del is then applied to make forecasts for all time horizons. This is considered optimal if the one-step-ahead model, including the distribution of the error term, is correctly specified. Nevertheless, it is well-known in the forecasting literature that this procedure may not be optimal in practice.

The papers in the literature review provide the theoretical background of how different time series may affect the economy over multiple time scales. Given the possibility that variables may affect the economy through various time spans, we likely need to include different variables into different-step-ahead forecasting models as the forecasting horizon increases because it is infeasible to accommodate all key variables corresponding to their own horizons in a traditional one- step-ahead model. The new assumption raises the question of whether the common practice of the VAR model using the same variables at all horizons is appropriate. It is straightforward to test this doubt without making any strong assumption.

My primary goal in this paper is to determine whether a model allowing different variables specific to a given horizon has a lower Schwarz Bayesian Information Criterion (SBIC) value than a model that does not. I also would like to gauge if a multiple-step-ahead model in which variables are selected corresponding to their horizons has a lower out-of-sample mean squared error (MSE) ratio by iterated forecasts than the standard VAR model. Does a model allowing different variables specific to the horizon have a lower out-of-sample MSE than a model which does not?

Furthermore, for a factor model, conventional methods focus on selecting the principal components from the front to be modeled. Nevertheless, they ignore the possibility that the selected principal components may vary as the corresponding horizon differs. I make use of 190 different time series to calculate the principal components and then select the optimal principal components to forecast a single time series, one by one, specific to a given horizon. I check whether a model allowing different principal components specific to the horizon has a lower out-of-sample MSE by direct forecasts than a model that does not. Do the principal components at the end have lower out-of-sample MSEs than the principal components in the front if we allow the horizon to differ? As I shall argue, I find my assumption more appealing than the conventional assumption that variables should be same for all horizons.

I contribute to the literature in four ways:

・ This paper constructs a novel framework providing a systematic way to select variables specific to the horizon, with fewer coefficients than a VAR model. I demonstrate that variables should be modeled specific to the horizon. Including all variables in a one-step-ahead model is not sufficient to resolve the question of the relative importance of different variables which may change as the horizon varies.

・ I also set the sample mean as a benchmark to judge the forecasting performance of VAR models and find that it is better to make forecasts by the sample mean than traditional VAR models with lags longer than one. I demonstrate that the one-step-ahead VAR model forecasts GDP poorly during recessions relative to the multi-step-ahead models I select. This in turn indicates that the model which allows variables specific to the horizon enhance the predictive ability of the VAR model using out-of-sample forecasts.

・ My results indicate that we should reselect principal components as the time horizon changes. The principal components from the front do not necessarily forecast better than the principal components from the end as the time horizon varies.

・ Finally, the selection results are done to see if the same problem plagues factor-augmented VAR (FAVAR) models. The FAVAR model is considered to include information on all variables using a few factors. My results suggest that since some of the principal components may be more important at some horizons than at others, we have to select the principal components in a FAVAR model specifically to the horizon.

A potential criticism of my approach is that I will arbitrarily select some variables by some criterion through computer programs. Since my primary focus is to demonstrate that the variables in forecasting models will change as the horizon changes, it is clearly the case that the variables are same for all time horizons if my selected results will be same for all horizons. Otherwise, if any variable changes with the time horizon, it is possible to show that the importance of variables may depend on the exact horizon. The selected forecasting models through computer programs need further regression analysis. I do not consider that this limitation to be overly problematic. I am rather interested in verifying the possibility that variables vary as horizon changes, as opposed to explaining it. In other words, I try to prove that we should build scale-wise models with variables specific to the horizon rather than actually build a theoretical model by computers in this paper.

Additionally, as Box (1979) [1] noted, “All models are false, but some are useful.” Stock and Watson (1999) [2] mention that even if the model is misspecified, it may still produce reasonable one-period-ahead forecasts. In this paper, I claim that forecasts derived by iterating forward multi-step-ahead projects with variables selected in the multi-step-ahead model may enable us to improve the forecast accuracy of some time series during recessions, even though these variables may be ignored by traditional one-step-ahead model analysis. Even though the omitted variables in the error term that affect the economy directly through other horizons may be correlated with the variables on the right-hand side (R.H.S.) of a model¹, I claim that my approach may be appropriate insofar as selecting forecasting models for recessions relative to the conventional models. The remainder of the paper is organized as follows. Section 2 presents a literature review in support of my assumption that variables in a model may not be the same for all horizons. Section 3 outlines my novel methodology and Section 4 conducts a small simulation experiment to determine the probability that my argument is spurious. Section 5 compares my results with the out-of-sample forecasts of an SVAR model, while section 6 analyzes the selected forecasting models with principal components. To provide further empirical evidence, I investigate the forecasts of US real GDP and the inflation rate during recessions and discuss the implications of the results. Concluding comments and directions for future research are given in Section 7. The Appendix summarizes the data sources.

Since Sims (1980) [4] , Doan et al. (1984) [5] , Littlerman (1986a) [6] , the VAR model has become a useful tool for making out-of-sample forecasts in macroeconomics, which approximately captures the coefficients of multiple variables in a one-step-ahead model and predicts the fluctuations of variables in the future. VAR models tend to suffer from over-parameterization and problematic predictions by caused by an excess of free insignificant parameters. Shrinkage methods have been proposed to resolve this problem of VAR modelling, like variable selection, factor models (Stock and Watson, 2005) [7] , FAVAR models (Bernanke et al., 2004) [8] and so on. For variable selection, traditional methods focus on which―and how many―variables to include in the model from the candidate variables nevertheless ignore the possibility that selected variables may vary as the corresponding horizon differs. The same problem may also plague factor models and FAVAR models. Models, especially for macroeconomic models, will always omit variables. The key, however, is knowing if the omitted variables are important and how they are going to impact our models. By selecting the variables specific to the horizon, I try to include the important variables in each horizon to decrease free insignificant parameters.

The assumption that variables should be the same for all horizons is in fact almost always subject to serious challenges such as variance decomposition evidence and so on. Despite this, the poor forecasting performance of VAR models has not been attributed to the characteristics of variable variation across horizons. For example, Friedman (1961) [9] advocates that for the eighteen non-war business cycles since 1870, monetary policy affects economic conditions only after a lag which is long and variable.

Likewise, Blanchard and Quah (1989) [10] appeal to an analogous argument regarding that some variables are more important at some horizons than at others. When they check the forecast error variance decompositions of output at multiple horizons, they find that the contribution of demand disturbances to output is above 80% before the 8^th forecast period, while it drops sharply to 39.3 after 40 periods. This indicates a decline of the contribution of demand disturbances in explaining the movements of the output. At the same time, the contribution of supply disturbances to output increases over time. They point out that demand disturbances have a hump-shaped effect on output, which disappears after about two years, while supply disturbances have a continually increasing effect on the output which reaching a plateau after five years. Their findings are consistent with my argument that the supply and demand shocks may play important roles in explaining output at different horizons. If we focus only on the short-horizon evidence, we may make use of only demand interventions to analyze in the model, and many variables affecting output persistently and strongly in some long horizons may be ignored.

Kilian (2009a) [11] initially identifies oil-specific demand shocks and oil supply shocks. He postulates that global oil production does not response to oil demand shocks contemporarily based on costs to adjusting production and anecdotal evidence on OPEC production decisions. Furthermore, his model imposes a delay restriction on feedback from fluctuations in the real price of oil to global real activity. Kilian rules out instantaneous feedback within the month. His delay restrictions advocate that not all variable will affect other variables in the economy immediately. Lippi and Nobili (2009) [12] implement a closely related approach to decompose oil demand and oil supply shocks. In their Table 3, they provide compelling evidence that the US aggregate demand shocks explains the largest share at the short horizons (1 - 6 months) and its role becomes smaller than the role of US aggregate supply shocks at all subsequent horizons. The variance decomposition part of an extensive number of studies on identifying different kinds of shocks explores the idea that the variables considered to play essential roles swing over different time horizons.

Cassou and Vázquez (2012) [13] contribute to the VAR literature that the well-known lead and lag patterns between output and inflation arise mostly over medium-term forecast horizons.

These papers provide evidence for my assumption that variables do not necessarily require a relationship through one-step ahead, which is the foundational principle behind the approach outlined in the following section.

In the usual approach to making multi-step-ahead forecasts, economists select a one-step-ahead VAR model and use the same VAR model to make forecasts multi-step ahead. Researchers typically proceed as if they are absolutely certain that the variables are same for all forecasting horizons while having no useful information about another perspective that the substantial contributions of a variable may change as horizon differs. Though the approach implementing the same variables for all horizons is extremely prevalent in the literature, this paper selects macroeconomic variables and lag lengths in the multiple-step-ahead VAR model using a criterion specific to the horizon, which is mostly disregarded in mainstream discussion.

Equation (1) displays one equation in my multiple-step-ahead VAR model. Each equation has a single variable on the left-hand side, denoted as y t + s . Other variables on the right hand side (R.H.S.) as explanatory variables, denoted as X t − i . Equation (1) regresses a multi-period-ahead value of a predicted variable on its past values and the past values of other explanatory variables selected by the lowest SBIC. This criterion is recommended by Diebold (2015) [14] as it is based on the full sample data. I employ the following forecast equation:

y t + s = α s + ∑ i = 1 p γ i s + 1 y t − i + ∑ i = 1 p β i s + 1 X t − i + ε t + s s ,   S = 0 , 1 , 2 , ⋯ , h (1)

where y t + s is the dependent variable we want to forecast s steps ahead for h different forecast horizons. y t − i is the i t h lag of the left-hand-side variable, X t − i is a vector of explanatory variables for each lag i, and the number of deterministic variables in X is N. P denotes the number of lags, with a limit of P ≤ 12 in this paper. α s is the constant in the s-step-ahead forecasting equation, β i s + 1 denotes the matrix of parameters corresponding to the i t h lag of N variables in X t − i , and ε t + s s is the forecast error term of this s-step-ahead equation.

I will use Equation (1) to reselect variables in X from all combinations of variables, and lag length by SBIC. The lag length P, variables in X, and the number of variables N need not be same across different horizons.

For instance, I have m variables: x 1 , x 2 , ⋯ , and x m . To select the variables and lag length in the model for s steps ahead, I can proceed in the following steps:

1) In step one, for the s-step-ahead forecasting model with one variable in X in Equation (1):

a) First, I regress the s-horizon-ahead value of the variable, y t + s , on its past value y t − 1 and the past value x 1, t − 1 , with the lag length equal to one. I estimate this equation to calculate the SBIC.

b) Second, I use the same variable x 1 , whereas change the lag length P in Equation (1) from one to two and so on, finally to twelve to get twelve SBICs in total.

c) Third, I use another variable x 2 , with lag length from one to twelve to get twelve SBICs. SBICs are then estimated. And the final SBICs with only one variable in X are estimated from regressions of y t + s on its past values, and the past values of the last variable x m , with the lag length from one to twelve. The total number of SBICs for only one variable in X is thus 12m.

2) In step two, for two variables in X, I regress y t + s on its past values and the past values of any combination from all combinations of two variables in X, with the lag length from one to twelve. I then get 6 m ! / ( m − 2 ) ! SBICs.

…

3) In step m, for all m variables in X, the last SBICs are calculated from regressions of y t + s on its past values and past values of all m variables, with a lag length P from 1 to 12.

After I collect all SBICs for all combinations of variables for Horizon s, I select the most favorable model by the lowest SBIC. Further, I reselect the variables and lag length by changing horizons. Finally, I can compare the SBIC outcomes for different time horizons and check if the selected variables change as the time horizon changes.

Briefly, the coefficients of the unselected variables are set to zero derived by data set automatically. This means that some variables may not have a substantial contribution in some horizons. I set the sample mean as the out-of-sample forecasting benchmark.

I use simulation to argue that under my assumption, we need to include different variables in the model corresponding to different horizons. It is a basic tenet that the only relevant information for output now is localized in the short run, and no valuable gain is obtained by instead directly incorporating long-horizon information and coefficients estimated from the long horizon models, and we can use the variables selected one-step ahead to mechanically infer the dynamics of the dependent variable through forward aggregation by these same variables for all horizons. The simulation process tries to show that if the data generation process (DGP) is such that two shocks affect the economy differently at different horizons, the SBIC of a model that allowing the variables to change at each horizon will be lower than when the variables do not change at each horizon. In such a case, mechanically generating the dynamics of the dependent variable through forward aggregation of the selected variables one-step ahead will miss the effect of the variables that affect the dependent variable over long horizons².

First, I assume that there are two types of shocks, each uncorrelated with the other. They affect the economy through different horizons. I interpret the disturbances that affect output y in horizon h = 0 : 6 as being demand disturbances, and those only affect output y in horizon h = 14 : 20 as supply disturbances. The effect of demand disturbances on y can change to be negative in h = 4 : 6 . Moreover, the supply disturbances are assumed to have a lower frequency than demand disturbances. The function is as following:

y t = 0.4 e D ,   t + 0.3 e D ,   t − 1 + 0.2 e D ,   t − 2 + 0.1 e D ,   t − 3 − 0.3 e D ,   t − 4 − 0.2 e D ,   t − 5 − 0.1 e D ,   t − 6 + 0.4 e S ,   t − 14 + 0.3 e S ,   t − 16 + 0.2 e S ,   t − 18 + 0.1 e S ,   t − 20 (2)

While e D ,   t , e S ,   t ∼ N ( 0 , 1 ) . I also postulate that the demand variable u t and the supply variable z t are affected by the demand shocks and the supply shocks, respectively:

u t = 0.4 e D ,   t + 0.2 e D ,   t − 1 (3)

z t = 0.4 e S ,   t + 0.2 e S ,   t − 1 (4)

To simulate data, I first draw 500 normally distributed random values for each type of shocks and use the above functions to calculate y t , u t , and z t . Then I select the variables and lag length ( ≤ 12 ) from all combinations of variables to forecast output y t using Equation (1) by SBIC in each horizon. Then I repeat the above process 1000 times and count the number of u , z , and both u and z which are selected in each horizon.

Table 1 shows the times of the selected variables to forecast y from all combinations of variables u and z by the lowest SBIC (denote by X in Equation (1)) at each horizon after performing 1000 simulations. The corresponding time horizon, the times of u , z , and both u and z which are selected at this horizon are listed in column 1 to 4, respectively. I set the selected lag length p for the selected variables less or equal to 12. For the results of Table 1, the demand