Almon-KL estimator for the distributed lag model

Abstract The Almon technique is widely used to estimate the parameters of the distributed lag model (DLM). The technique suffers a setback from the challenge of multicollinearity because the explanatory variables and their lagged values are often correlated. The Almon-Ridge estimator (A-RE) and Almon-Liu estimator (A-LE) were introduced as alternative estimators for efficient modelling. We developed a new method of estimating the coefficients of the DLM using the Almon-KL estimator (A-KLE). A-KLE dominates the other estimators considered in this study via theoretical findings, simulation design and two numerical examples. The estimators’ performance was compared using the mean squared error.


Introduction
Distributed lag model (DLM) is a model that is commonly adopted to predict the current values of a response variable based on both the current values of a regressor (explanatory variable) and its lagged values (Cromwell, Hannan, Labys, & Terraza, 1994;Judge, Griffiths, Hill, & Lee, 1980). For instance, the effect of public investment such as road and highways construction on growth in the gross national product (GNP) will show up with a lag, and this effect will likely linger on for some years (Maddala, 1974). Under certain assumptions, the ordinary least squares estimator (OLSE) is the easiest method of estimating the distributed lags' parameters. The assumptions include assuming a fixed maximum lag; the error should be normally distributed and unrestricted relationship among the lagged explanators. However, multicollinearity arises from the relationship between the explanatory variables and their lagged explanators, resulting in a large variance of the coefficient estimates. The OLSE becomes inefficient when there is multicollinearity. Also, the method suffers a breakdown when the number of observations does not sufficiently exceed the number of lags (Almon, 1965;Fisher, 1937;Frost, 1975;Maddala, 1974). Almon technique proposed by Almon (Fisher, 1937) is widely used to estimate the parameters of DLM because the methods reduced the effect of multicollinearity (Fair & Jaffee, 1971;Majid, Aslam, Altaf, & Amanullah, 2019). The assumption under the Almon lag model is that there must be a finite lag length and the lag weight approximated by a polynomial of suitable degree.
The Almon estimator (AE) is the best linear unbiased estimator (BLUE) for the DLMs (Vinod & Ullah, 1981). Therefore, this estimator has widespread usage in applied econometrics due to its ease of estimation. However, the estimator still suffers a breakdown when there is a high correlation among the explanatory variables. Biased estimators were developed as alternatives to the OLS estimator in the linear regression model (LRM) (Baye & Parker, 1984;Hoerl & Kennard, 1970;Kaçıranlar & Sakallıo glu, 2001;Li & Yang, 2012;Liu, 1993;Lukman, Ayinde, Kibria, & Adewuyi, 2020;Massy, 1965;Swindel, 1976). Some of these estimators have been adapted to AE to mitigate the effect of multicollinearity.
The ridge estimator was suggested as an alternative to the AE (Chanda & Maddala, 1984;Lindley & Smith, 1972;Lukman, Ayinde, Binuomote, & Onate, 2019;Maddala, 1974;Vinod & Ullah, 1981;Yeo & Trivedi, 2009). The limitation of the ridge regression estimator for the DLMs was pointed out in the following study (Lindley & Smith, 1972;Maddala, 1974;Yeo & Trivedi, 2009). According to Maddala (1974) and Yeo and Trivedi (2009), the empirical findings do not fulfil expectations despite adopting different ridge parameters, k. Alternative estimators include the Almon modified ridge and Liu estimators (G€ ultay & Kac¸ı ranlar, 2015). € Ozbay & Kac¸ıranlar (2017) introduced the Almon two-parameter estimator to deal with the problem of multicollinearity for the DLM. The estimator extends the two-parameter estimation procedure by € Ozkale and Kaçıranlar (2007) to the DLM. The Almon two-parameter ridge was also developed to handle multicollinearity in DLM (Lipovetsky, 2006;Lipovetsky & Conklin, 2005;Ozbay, 2019). Kibria and Lukman (2020) developed the Kibria-Lukman estimator (KLE) to cushion the effect of multicollinearity in the LRM. The estimator gain advantage over the ridge and the Liu estimator. Thus, the focus of this study is to develop the Almon-KL estimator (A-KLE) based on KLE.
The other part of the article is structured as follows: we introduced the DLM, its estimation methods, and the proposed estimator in Section 2. Section 3 deals with the theoretical comparisons and the choice of biasing parameters. We examined the estimators' performance using a simulation study and two examples in Sections 4 and 5, respectively. Finally, we provide a concluding remark.

Distributed lag model (DLM) and method of estimation
The finite DLM is defined as where b i are the current and lagged coefficients of x t , y t denotes the t-th observation on the response variable y, x tÀp denotes the (t-p)th observation on the explanatory variable x, u t is the disturbance term corresponding to the t-th observation and is assumed to be IN(0, r 2 ) and p denotes the lag length. Model (2.1) can be written in matrix form as . . . . . .
The OLS estimator of b is BLUE when there is no violation of the model assumptions (Lukman et al., 2019). As earlier mentioned, there is high tendency of multicollinearity among the regressors since lags of the same explanatory variable appear in the model (G€ uler, G€ ultay, & Kaçiranlar, 2015;G€ ultay & Kaçıranlar, 2015). It becomes difficult to estimate with the conventional OLS when the lag length (p) is known and large (G€ uler et al., 2015;G€ ultay & Kaçıranlar, 2015;Ozbay, 2019;€ Ozbay & Kaçıranlar, 2017). The Almon polynomial lag distribution was recommended as a replacement to the OLS estimator (Almon, 1965). The Almon polynomial lag is defined as The above equation can simply be written as are a : ðs þ 1Þ Â 1 vector and A : ðp þ 1Þ Â ðr þ 1Þ matrix, respectively. The ranks of matrices X and A are assumed to be ðp þ 1Þ<ðTÀpÞ and ðs þ 1Þ<ðp þ 1Þ, respectively. If s<p then the rank of A is s þ 1. We substitute (2.5) into (2.2) and obtained the new equation as, where Z ¼ XA: The OLS estimator of (2.6) becomes the AE in (2.7):â Equation (2.8) is referred to as AE of b which is still BLUE. A notable problem with this technique centered on the choice of lag length and the degree of the polynomial, respectively because the two are unknown in practice. One of the suggested means of selecting the lag length is to select a reasonable lag and check if it fit the model (2.1) using any of the following criteria: Akaike information criterion (AIC), Bayesian information criterion (BIC) and adjusted coefficient of determination ð R 2 Þ (Davidson & MacKinnon, 1993). The ridge estimator was suggested as an alternative approach due to the limitation of the AE. The whereâ k ¼ ðZ 0 Z þ kIÞ À1 Z 0 y: € Ozbay and Toker (2021) examined the predictive performance of A-RE.
According to G€ ultay & Kaçıranlar (2015), the Almon-Liu estimator (A-LE) is defined as: The Liu estimator overcomes the limitation in the ridge estimator for the LRM. Recently, Kibria and Lukman (2020) developed the KL estimator (KLE) to mitigate multicollinearity in the LRM. The estimator is a modification of the Liu estimator; it gained an advantage over the Liu estimator using the mean squared error. The bias of KLE is two times lower than the bias of the ridge estimator. Thus, the KL estimator performs better than the ridge and the Liu estimators. Following this merit, the estimator is adapted to the AE to handle multicollinearity in this study. The A-KLE is defined aŝ :, e sþ1 Þ such that e 1 ! e 2 ! ::: ! e sþ1 are the ordered eigenvalues of Z 0 Z and Q is the matrix whose columns are the eigenvectors of Z 0 Z: and the canonical form of the estimators are as follows: The statistical properties of A-KLE are as follows: The bias of the A-KLE is defined as: The variance of the A-KLE is defined as: Therefore, the mean square error matrix (MSEM) and the scalar mean square error (SMSE) of A-KLE, respectively, are and (2.19)

Theoretical comparisons among the estimators
In this section, we adopted Lemma 3.1 to evaluate the estimators' performance.

Comparison betweenâ
It is noted that E À1 ÀE k ðEÀkIÞE À1 E k ðEÀkIÞ is nonnegative (nn) because e i þ k>e i Àk for k>0:

Comparison betweenâ k andâ KL
Theorem 3.2. The estimatorâ KL is preferred toâ k in the MSEM sense, if and only if, b 0 ½r 2 ðE k EE k ÀE k ðEÀkIÞ E À1 E k ðEÀkIÞÞ þb 0 1 b 1 À1 b<1 where b ¼ À2kE k a and b 1 ¼ kE k a: Proof.
It is noted that E k EE k ÀE k ðEÀkIÞE À1 E k ðEÀkIÞ is non-negative because e 2 i Àðe i ÀkÞ 2 >0 for k<2e i :
(4.1) ). An observation of T-p ¼ 60 and 100 with the lag length of 8 and 16 was evaluated. The experiment is repeated 2000 times using the RStudio programming language (R Core Team, 2020). The model (4.1) is transformed back to the Almon model (2.6) after generating the observations. The mean squared error is obtained as follows: whereb ij denotes the estimate of the ith parameter in jth replication and b i is the true parameter values. The simulated MSE values are presented in Tables 1  and 2 for lag length are 8 and 16, respectively.
From Tables 1 and 2, we observed the following trend about the mean squared error. The mean squared error increases as the following increases: level of multicollinearity, lag length, and the error variance. The mean squared error decreases as the sample size increase. This trend is more feasible considering Figure 1. The results show that AE suffers setback when the regressors are correlated. The biased estimators provide a more consistent estimate when there is multicollinearity. However, the A-KLE produces the best result by producing smaller mean squared error among all the estimators under study.

Applications
To illustrate the theoretical findings of the paper, we consider the two real datasets.

Application example I
The Almon dataset is employed to illustrate the performances of each of the estimators (Almon, 1965). These data are quarterly data that spanned from 1953 to 1967 with expenditures as the response variable and appropriations as the independent variable. Following G€ uler et al. (2015), the lag length is taken to be 8 and the lag weight approximated by a polynomial of order 2. The eigenvalues of Z 0 Z are computed to be 2.388929e þ 13, 1.628323e þ 09, 4.312742e þ 07, and 4.782067e þ 00, respectively. The condition index of CI ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi maxðe i Þ=minðe i Þ p is determined to be 2235083, which revealed the presence of severe multicollinearity among the explanatory variable and its lagged variables. The library dLagM in RStudio is used to analyse the data. The biasing parameter for A-RE and A-KLE is computed using: The biasing parameter for the A-LE is obtained using the following:    The regression parameter before transformation is presented in Table 3 as follows: By canonical transformation, the 4 Â 4 matrixQ is the matrix of the eigenvectors, E is a 4 Â 4 diagonal matrix of eigenvalues of Z 0 Z such that Z 0 Z ¼ QEQ 0 : The Almon-estimators of a are presented in Table 4. The result in Table 4 supports the fact that the Almon-estimator suffers set back when there is multicollinearity. The regression coefficient of the biased estimators appears very similar. The A-KLEr has the lowest mean squared error followed by the A-RE.

Application example II
This dataset is composed of monthly global mean sea level (GMSL) (compared to 1993-2008 average) series by CSIRO, land ocean temperature anomalies (1951-1980 as a baseline period) by GISS, NASA, and monthly Southern Oscillation Index (SOI) by Australian Government Bureau of Meteorology (BOM) between July 1885 and June 2013 (Church & White, 2011;Demirhan, 2020). In this study, the response and the explanatory variables were chosen to be GMSL and land ocean temperature anomalies, respectively. The dataset is available in the R package data (seaLevelTempSOI). More detail on the data description is available in the study of Demirhan (2020). The lag length is taken to be 4 and the lag weight approximated by a polynomial of order 2 (Demirhan, 2020). The eigenvalues of Z 0 Z are computed to be 6607.86, 1590.99, 67.12 and 6.96, respectively. The condition number is 948.82 which shows that the explanatory variable and its lagged variables are correlated. The regression parameter before canonical transformation is presented in Table 5.
By canonical transformation, the Almon-estimators of a are in Table 6. From Table 6, the Almon-estimator for the second lag has a positive sign as opposed a negative sign by other estimators. As mentioned earlier, one of the effects of multicollinearity on the AE is that the coefficients can occasionally exhibit a wrong sign. The results show that the new estimator has the lowest mean squared error. This result agrees with the theoretical and simulation result.

Some concluding remarks
The DLM is often adopted to predict the current values of a response variable based on both the current values of a regressor and its lagged values. Multicollinearity arises from the relationship between the regressors and their lagged explanators. The AE is popularly used to estimate the parameters of the models. However, the estimator suffers a setback when there is multicollinearity. The A-RE and A-LE estimators were suggested as alternative techniques. In this study, we developed the A-KLE to mitigate the threat of multicollinearity for the DLM. We theoretically proved that A-KLE dominates AE, A-RE and A-LE. The theoretical findings were supported by the results of the simulation study and two numerical examples.