Two-Step Optimal Prediction Under Phillips Triangular Cointegrated System

1 Introduction

And God shall wipe away all tears from their eyes; and there shall be no more death, neither sorrow, nor crying, neither shall there be any more pain: for the former things are passed away. REVELATION 21-1.

The problem of using cointegration information for forecasting has focused on the role of cointegration errors or error correction terms in predicting differences in cointegrated variables. Classical examples in this area include Engle and Yoo (1987), Christoffersen and Diebold (1998), and Elliott (2006). However, when the focus is on predicting the level of a cointegrated variable, the long-run cointegration equilibrium becomes important, which has received little emphasis.^[1] An exception is Kim (2023), who addresses this issue in the triangular form of Phillips (1991) for cointegration models.

In particular, Kim (2023) introduced the best linear predictor (BLP) with an asymptotic minimum mean squared forecasting error (MSFE) among the linear predictors of variables in cointegrated systems. Kim (2023) showed that if the autocorrelation coefficient of the cointegration error between the prediction time and predicted targeting time is greater than ½, the BLP is deduced from the random walk model. In other cases, the BLP is deduced from the cointegration model. Under this scheme, Kim (2023) suggested a switching predictor that automatically selects a random walk or cointegration model according to the size of the estimated autocorrelation coefficient. He showed that the BLP has a weighted average form of the predictors using the random walk and cointegration models and has the lowest MSFE among the linear form predictors using the variables in cointegrated systems or their lagged variables. Here, there is a difference in O _p (T²) between the BLP and the other linear-form predictors. Note that the BLPs may differ in O _p (1) depending on the weighting coefficient; however, Kim (2023) did not suggest a weighting coefficient that minimizes MSFE.

Under these circumstances, to improve the forecasting efficiency of the BLP, we propose the optimal best linear predictor (OBLP), which is deduced from a two-step optimal forecasting method for non-stationary level variables cointegrated with the fundamental variables. To do this, the cointegration equilibrium is estimated in the first step. The difference between the cointegration equilibrium and the other predicted variables is optimally forecasted in the second step, with conditional expectations estimated by the lagged fundamental differences and cointegration errors and summed with the cointegration equilibrium. We show that the OBLP has the lowest MSFE among linear forecasting methods, such as random walk, cointegration, and augmented error correction models. In the second step, the cointegration error correction model is converted into a vector autoregression (VAR) model consisting of the cointegration error and the difference in the fundamental variables and is used to estimate conditional expectations.

Note that following Engle and Yoo (1987), Christoffersen and Diebold (1998), and others, Elliott (2006, p. 584, Eq. 11) addressed the problem of optimal forecasting of co-integrated differenced variables in a bivariate VAR(1) model. An OBLP can equivalently be deduced by adding Elliott’s (2006) predictor to the forecast baseline-level variable; however, an OBLP has not yet been provided for the general VAR(q) model.

The remainder of this paper is organized as follows. Section 2 introduces the optimal BLP and Section 3 discusses the OBLP estimation. Section 4 provides the Monte Carlo simulation results, and Section 5 presents an application to the prediction of United States’ GDP and consumption. Finally, Section 6 concludes the paper.

2 Derivation of the OBLP

First, we assume that the r × 1-vector y _t and the k × 1-variable x _t explaining it are jointly represented by a VAR model; that is, we consider the ℓ   ≡ r + k -dimensional and integrated of order one VAR(p) process of Y _t given by

or

where Y t ≡ x t ′ , y t ′ ′ , Π ̄ = ∑ i = 1 p Π i , Φ = Π ̄ − I ℓ , Φ i = − ∑ j = i + 1 p Π j and ε _t is an ℓ × 1 vector of an independently and identically distributed (i.i.d henceforth) disturbance term with a finite variance Σ > 0, where I ℓ denotes an ℓ -dimensional identity matrix and Δ Y t ≡ Y t − Y t − 1 .

Further, we assume the cointegration of Model (1) (e.g., Johansen 1991) as follows:

Note that Model (2) may be written as an error correction model (ECM) as

under Assumption 1, where z _t  = β′Y _t .

Now, we transform Model (3) into a stationary VAR model of the I(0) variables Δx _t and z _t . To obtain this stationary VAR representation, we first define a non-singular square matrix as follows:

It should be noted that the lower triangular matrix N transforms the VAR variable Y t = x t ′ , y t ′ ′ into the variable w _t of x _t and cointegration error z _t .

Following Kim (2012, 2018), we multiply the above matrix N on the left-hand side of Model (1) and modify the VAR coefficients to obtain the following VAR model of the purely stationary variable w Δ t ℓ × 1 = Δ x t ′ , z t ′ ′ :

where e _t  = N × ε _t , or a state space form:

where W Δ t ℓ p × 1 = w Δ t w Δ t − 1 ⋮ w Δ t − p + 1 , Ψ 0 = ψ 0 0 ⋮ 0 , Ψ ℓ p × ℓ p = ψ 1 ψ 2 ⋯ ψ p − 1 ψ p I ℓ ⋯ 0 0 ⋮ I ℓ ⋮ ⋱ 0 I ℓ 0 and e t = e t 0 ⋮ 0 .

Note that the columns of ψ _p from the first to k-th are imposed as zero vectors/matrices, following Kim (2012, Theorem 3.2); thus, Δx_t-p does not appear in Equation (4).^[2]

We define two selection matrices, M_a,b≡(I _a ,0_a×b) and M ̄ a , b ≡ 0 a × b , I a . Now, equation (4) can be regarded as a Phillips (1991) triangular representation of a cointegrated system, as follows:

and

for t = 1, 2,…, T, where μ = M_k,rψ₀, δ = M ̄ r , k ψ 0 , u t = M k , r ∑ i = 1 p ψ i w Δ t − i + e t and z t = M ̄ r , k ∑ i = 1 p ψ i w Δ t − i + e t .

At time t, we aim to predict the variables y_it+h for 1 ≤ i ≤ r and h ∈ Z⁺, where Z⁺ denotes a set of positive integers, and y t ≡ y 1 t , y 2 t , . . . , y r t ′ . Let δ ^ , γ ^ ′ be an OLS (ordinary least square) estimator of (δ,γ′).

Furthermore, we assume the following standard regularity conditions, as in Kim (2023):

However, the conditional expectation E _t (y_t+h), which is the optimal predictor when y _t is I(0), is not defined in general because there are no finite moments of y _t when y _t is I(1). Therefore, OBLP for y_t+h is defined from (12) as the long-run cointegration equilibrium of y _t after adding the conditional expectation of q_t+h, which is I(0), as follows:

We now derive the difference in MSFE between the LP and OBLP as follows:

According to Theorem 1, the LP has a larger MSFE than the OBLP owing to a positive definite matrix of size O _p (T²). Next, the difference in the MSFE between the BLP and OBLP is given as

Next, the OBLP for y_it+h is given by, which has the minimum MSFE among the BLPs. To demonstrate this, we first define two predictors of y_it+h.

and

where m _i = (0_(i-1)×1,1,0_(r-i)×1) and c _t is an arbitrary I(0) real variable.

For instance, F t B L P i is a predictor of y_it+h using the cointegration error z _it ^[7] in a VAR(1) model that is conformable with y_it+h. Note that this is not a predictor deduced from the OBLP of y_t+h, as in (13), using all cointegration error vectors z _t .

Accordingly, the optimality of the predictor (14) for y_it+h is given by

3 Estimation of the OBLP

In this section, we introduce a consistent estimator of the OBLP and demonstrate that the estimated OBLP asymptotically has the same MSFE as the OBLP. To do so, we first rewrite the last three terms on the right-hand side of Equation (12) as follows:

because q_t+h = E _t (q_t+h) + ε_t,h from definition.

Then, we define the estimated OBLP as

where the coefficients in (16) are estimated using OLS as follows:

where z ^ t ≡ y t − δ ^ − γ ^ ′ x t , w ^ Δ t = Δ x t ′ , z ^ t ′ ′ ,

Now note that

We now show that the suggested OBLP estimator has the same MSFE as the OBLP in (15), as follows:

Finally, the OBLP for y_it+h is given by

The OBLP estimated in this manner can also be shown to have the same predictive efficiency as Theorem 3 based on the consistency of the OLS estimates.

In the example below, we show how the OBLP presented earlier is applied to the VAR(1) model.

4 Monte Carlo Simulation Results

We conducted a Monte Carlo experiment^[10] to verify the small-sample properties of the proposed predictors. The basic simulation model used has the following triangular form:^[11]

and

for t = 1, 2, . . ., 100 and u t , ε t ′ ∼ IID N 0 , I 4 , where y t = y 1 t , y 2 t ′ . The parameters are set as follows:

γ ′ = 0.5 0.1 0.1 0.5 , Ψ = λ ρ ρ λ , λ = 0.5 or 0.7, ρ = 0.1,…, 0.4, δ = 0.1 0.1 and μ = 0.1 0.1 or 0.5 0.5 , respectively. Here, ρ denotes the correlation of the different co-integration errors.

Then, y_1t+h is predicted at time t using 100 samples, where h = 1, 2, . . ., 20. The MSFEs are calculated for the five predictors; RWP, CIP, OBLP, restricted OBLP using only the cointegration error of the predicted variable y_1t ( OBLP1) and ECM predictor (ECMP).^[12] We also add a deterministic trend to the predictors to obtain

1. RWP: 1 , 0 ′ × y t + h γ ′ μ

2. CIP: 1 , 0 ′ × δ + γ ′ x t + h γ ′ μ

3. ECMP: 0 , 0 , 1 , 0 ′ Y t + E Y t + h − Y t z t E z t z t ′ − 1 z t + 1 , 0 ′ × h γ ′ μ

Subsequently, the MSFEs are computed as the mean of the samples from 10,000 repetitions of the aforementioned experiments.

The simulation results in Appendix A confirm the theoretical expectation; that is, when the prediction period h is small, the OBLP has the best MSFE among the predictors in terms of prediction stability and efficiency. In Appendix B, the ratio of the relative size of the MSFE of predictor b to that of the OBLP is calculated as

and plotted it on a graph with the forecast horizon (1–20) on the x-axis.

From the calculations, we obtained the following results. First, the variation in the cointegration matrix γ does not significantly change the forecast results. Second, an increase in ρ leads to an increase in the forecasting efficiency of OBLP compared with that of OBLP1, which seems to be because the increased forecasting efficiency of OBLP uses additional cointegration errors from other equations in the forecast.

Third, the CIP shows a much lower forecast efficiency than the OBLP for short-term forecasts but slightly better efficiency than the OBLP as the forecast horizon increases. However, as the error correction process slows (i.e. as λ increases), the decrease in the MSFE of the CIP relative to that of the OBLP is delayed as the forecast period increases.

Finally, the RWP is inferior to the OBLP over the period, but converges to the OBLP as the forecast horizon increases, whereas the ECMP is less efficient than the OBLP as the forecast horizon increases. This phenomenon is further exacerbated as μ, which represents the magnitude of the deterministic trend, increases.

5 Application to the United States GDP and Consumption Prediction

In this section, we conduct out-of-sample forecasts for US GDP and consumption using the predictors suggested in Section 4. We compare the forecast performance with the MSFE calculated using h-period (1, 2,…, 20)-ahead estimated forecast errors. The data used have a quarterly frequency that extends from Q3 1976 to Q3 2023.^[13] Therefore, the analysis of the out-of-sample predictive performance of the proposed model consists of forecasting US GDP and consumption for each quarter from Q1 2018 to Q3 2023 using data from Q3 1976 to Q4 2017.

The data source is the United States Federal Reserve Board at St. Louis FRED. The cointegration fundamentals initially considered for US GDP and consumption are interest rate term spread, net exports, and government expenditure. All variables, except interest rate term spread and net exports, are log-transformed.

Before proceeding, we conduct augmented Dickey–Fuller (ADF) and Elliott–Rothenberg–Stock (ERS) point optimal tests to check the unit root of the variables considered. Table 1 presents the unit root test results. The ADF test results show that the null hypothesis (i.e. that the variable has a unit root) is not rejected at the 1 % significance level when the test equation does not include a trend or intercept term. The results of the ERS test show that the null hypothesis is not rejected at the 1 % significance level when the test equation includes a trend or an intercept term.

Therefore, although this is somewhat restrictive for GDP and consumption in the added trend or intercept term cases, we assume that all variables have unit roots and proceed with the following analysis:

We then conduct Johansen cointegration tests using a VAR model to check whether a cointegration vector exists in the VAR model. We set the lag length of the VAR model to 1, based on the most parsimonious Schwarz information criterion. The Johansen test results, shown in Table 2, indicate that the trace and maximum eigenvalue tests jointly indicate one cointegrating equation at the level of 0.05.

Next, we estimate the forecasting model presented in Section 4 and calculate the forecast errors, as shown in Appendix C. In Figure 1, the ratio of the relative size of the MSFE of predictor b to the OBLP is calculated as follows:

and plotted it on a graph with the forecast horizon (1–20) on the x-axis. The prediction errors for GDP in Figure 1 show that OBLP1 outperforms the other predictors for most forecast horizons when evaluated based on its forecasting efficiency and stability. The CIP is the strongest in long-run forecasting, but it shows a very large absolute value of forecast error in the short-term horizon.

Figure 1:

MSFE ratio relative to OBLP. Note: The ratio of the relative size of the MSFE of a predictor b to the OBLP is calculated as M S F E b − M S F E O B L P M S F E O B L P .

The forecasting and estimation results are generally consistent with the macroeconomic theory. First, OBLP1 has the best forecasting results when using the interest rate term spread, government expenditure, and net exports as co-integrating explanatory variables for GDP (or consumption).^[14] For example, adding M1 and consumption to the GDP forecast or excluding government expenditure leads to worse forecasting results.^[15] This is likely because M1 has a low direct correlation with GDP, or it may be because consumption already has redundant information about GDP forecasting that interest rate term spread, government expenditure and net exports already have, and therefore does not contribute much to the forecast.

However, changing the interest rate term spreads from the 10-year treasury constant maturity minus the 2-year treasury constant maturity to the 10-year treasury constant maturity minus the federal fund rate (FFR) seems to reduce the forecasting efficiency of the OBLP because the 2-year treasury constant maturity interest rate reflects the investment securities market conditions more closely than the FFR.

We also find that the single cointegration vector OBLP1 yields better forecasting results than the OBLP, with the two cointegration vectors of GDP and consumption as dependent variables. This reflects the lower correlation between errors in the cointegration of GDP and consumption, suggesting that the error correction process mechanisms for GDP and consumption are different. In addition, the forecasts of the CIP and OBLP class models tend to converge as the forecast horizon h increases.

6 Conclusions

This study proposes the OBLP, which is deduced from a two-step optimal forecasting method for non-stationary-level variables cointegrated with fundamental variables. For this, the cointegration equilibrium is estimated in the first step. The difference between the cointegration equilibrium and other predicted variables is optimally forecasted in the second step, with conditional expectations estimated by the lagged fundamental differences and cointegration errors, and summed with the cointegration equilibrium. We show that the OBLP has the lowest MSFE among the linear forecasting methods of random walk, cointegration, and augmented error correction models. In the second step, the cointegration error correction model is converted into a VAR model consisting of the cointegration error and the difference in the fundamental variables and is used to estimate conditional expectations. In the simulation results, we compare the other predictors with the OBLP, and our forecast results for the US GDP and consumption applying the OBLP support the theoretical predictions of the forecasting efficiency of the OBLP.

Finally, it would be interesting to apply the predictions using the OBLP to other macroeconomic variables, such as interest rates, stock prices, and exchange rates, in an empirical analysis.

Proof of Theorems