The Failure of Orthogonality under Nonstationarity: Should We Care About It?

Abstract

We consider two well-known facts in econometrics: (i) the failure of the orthogonality assumption (i.e., no independence between the regressors and the error term), which implies biased and inconsistent Least Squares (LS) estimates and (ii) the consequences of using nonstationary variables, acknowledged since the seventies; LS might yield spurious estimates when the variables do have a trend component, whether stochastic or deterministic. In this work, an optimistic corollary is provided: it is proven that the LS regression, employed in nonstationary and cointegrated variables where the orthogonality assumption is not satisfied, provides estimates that converge to their true values. Monte Carlo evidence suggests that this property is maintained in samples of a practical size.

1. Introduction

Two well-known facts lie behind this work: (i) the behavior of LS estimates whenever variables are nonstationary and (ii) the failure of the orthogonality assumption between independent variables and the error term, also in an LS regression. (1)The reappraisal of the impact of unit roots in time-series observations, initiated in the late seventies, had profound consequences for modern econometrics. It became clear that (i) insufficient attention was being paid to trending mechanisms and (ii) most macroeconomic variables are probably nonstationary; such an appraisal gave rise to an extraordinary development that substantially modified the way empirical studies in time-series econometrics are carried out. Research into nonstationarity has advanced significantly since it was reassessed in several important papers, such as those of [1–5]. (2)The orthogonality problem constitutes another significant research program in econometrics; its formal seed can be traced back to [6], where a proposal to solve the identification problem is made in the estimation of demand and supply curves (see [7]). Typically, in textbooks, the method of Instrumental Variables (IVs) is proposed as a solution to the problem of simultaneous equations and, broadly speaking, whenever there is no independence between the error term and the regressors, that is, when the orthogonality assumption is not satisfied.

This paper aims to study the consequences of using nonstationary variables in an LS regression when the regressor is related to the error term; this is done in a simple regression framework. The specification under particular scrutiny is𝑦_𝑡=𝛼+𝛽𝑥_𝑡+𝑢_𝑡.(1.1) To the best of our knowledge, the asymptotics—and the finite-sample properties—of the combination of nonstationarity, nonorthogonality between 𝑥_𝑡 and 𝑢_𝑦,𝑡, and LS estimates, have been scarcely studied (but see [8]). That said, we acknowledge that there are several comprehensive studies concerning the use of IV in the presence of nonstationarity [9, 10], for example, studied the asymptotics as well as the finite-sample properties of the IV estimator in the context of a cointegrated relationship, and proved that even spurious instruments (i.e., 𝐼(1) instruments not structurally related to the regressors) provide consistent estimates. Phillips [11] proved that, when there is no structural relationship between the regressand and a single regressor, that is, when there is no cointegration between 𝑦 and 𝑥, the use of spurious instruments does not prevent the phenomenon (this is a simple extension of [5]). We derive the asymptotic behavior of LS estimates, where the data generating processes (DGPs) consist of two cointegrated variables in which the regressor bears a relationship with the error term. In this case, LS provide consistent estimates. Additionally, some Monte Carlo evidence is presented to account for the adequacy of asymptotic results in finite samples. In other words, LS estimates of the true DGP parameters, 𝜇_𝑦 and 𝛽_𝑦 (see (2.4) in the next section), do not require the information on the parameters of 𝑥, that is, 𝑥_𝑡 is weakly exogenous for the estimation of 𝜇_𝑦 and 𝛽_𝑦 as defined by [12].

2. Relevant DGPs

This work aims to study the asymptotic properties of LS estimates when neither the orthogonality nor the stationarity assumptions are satisfied. Our approach is twofold: we assume (i) the variable 𝑥 is statistically related to the innovations of 𝑦, as in the problem of independent variables measured with error and (ii) the DGPs of both variables are interdependent, as in the problem of simultaneity. All the cases studied consider nonstationary and cointegrated variables (DGP (2.1) is included because it eases the comprehension of the paper)𝑥_𝑡=𝜇_𝑥+𝑢_𝑥,𝑡,(2.1)𝑥_𝑡=𝜇_𝑥+𝑥_𝑡−1+𝑢_𝑥,𝑡,(2.2)𝑥_𝑡=𝑋₀+𝜇_𝑥+𝜌_𝑦1𝑡+𝜉_{𝑥,𝑡−1}+𝜌_𝑦2𝜉_𝑦,t−1,(2.3)𝑦_𝑡=𝜇_𝑦+𝛽_𝑦𝑥_𝑡+𝑢_𝑦,𝑡+𝜌_𝑥𝑢_𝑥,𝑡_{innovations},(2.4) where 𝑢_𝑧,𝑡, for 𝑧=𝑥,𝑦, are independent white noises with zero-mean and constant variance 𝜎²_𝑧, 𝜉_𝑧𝑡=∑^𝑡_𝑖=0𝑢_𝑧𝑖 and 𝑍₀ is an initial condition. We may relax the assumptions made for the innovations; for example, we could force them to obey the general level conditions in [5, Assumption  1]. Nevertheless, although the asymptotic results would still hold in this case, our primary target concerns the problem of orthogonality between the regressor and the error term, not those of autocorrelation or heteroskedasticity. These DGPs allow for an interesting variety of cases (note that the asymptotics of the LS estimates when 𝑥 and 𝑦 have been independently generated by any of first three DGPs can be found, e.g., in [13]; notwithstanding, the authors can provide these cases as mathematica code upon request).(1)Bookcase no. 1: DGP of 𝑥 is (2.1) and DGP of 𝑦 is (2.4) with 𝜌_𝑥=0. When the variables are generated in this manner, we fulfill the classical assumptions made in most basic econometrics textbooks. The variables are stationary, the innovations are homoskedastic and independent, and so forth. It is straightforward to show that: 𝛼̂→^𝑝𝜇_𝑦, 𝛽̂‌→^𝑝𝛽_𝑦 and 𝜎̂→^2𝑝𝜎²_𝑦. (2)Bookcase no. 2: DGP of 𝑥 is (2.1) and DGP of 𝑦 is (2.4) with 𝜌_𝑥≠0. These DGPs also represent a typical example of a problem of orthogonality in most basic econometrics textbooks. Although the variables are stationary and the innovations are homoskedastic and independent, the explanatory variable is related to the innovations of 𝑦. It is well known that the estimates do not converge to their true value. In particular, it is straightforward to show that: 𝛼̂→^𝑝𝜇_𝑦−𝜇_𝑥𝜌_𝑥, 𝛽̂‌→^𝑝𝛽_𝑦+𝜌_𝑥 and 𝜎̂→^2𝑝𝜎²_𝑦. (3)Bookcase no. 3: DGP of 𝑥 is (2.2) and DGP of 𝑦 is (2.4) with 𝜌_𝑥=0. These DGPs allow the relationship between 𝑥 and 𝑦 to be cointegrated à la [14]. Once again, asymptotic results have been known for a long time, obtaining these does not entail any particular difficulty: 𝛼̂→^𝑝𝜇_𝑦, 𝛽̂‌→^𝑝𝛽_𝑦, 𝜎̂→^2𝑝𝜎²_𝑦 and 1−𝑅²=𝑂_𝑝(𝑇⁻²). (4)Nonstationarity and non-orthogonality case no. 1: DGP of 𝑥 is (2.2) and DGP of 𝑦 is (2.4). Notwithstanding, the obvious problem of orthogonality between 𝑥 and the error term, the variables remain cointegrated. The artifact employed to induce the orthogonality problem can be considered as, for example, measurement errors in the explanatory variable. One should expect that, in the presence of this problem, estimates would not converge to their true value. We prove below that, contrary to expectations, this is not the case. (5)Nonstationarity and non-orthogonality case no. 2: DGP of 𝑥 is (2.3) and DGP of 𝑦 is (2.4). As in the previous case, we have a cointegrated relationship between 𝑥 and 𝑦, only in this case, the problem of orthogonality between the regressor and the error term is even more explicit; the artifact employed to induce the orthogonality problem can be related to the typical simultaneous equations case. We also prove below that Least Squares (LS) provide consistent estimates.

The common belief as regards the last two cases is that the failure of the orthogonality assumption induces LS to generate inconsistent estimates, even in a cointegrated relationship. In fact, when the variables are generated as in (2.2)–(2.4), the estimates of the parameter converge to their true value (note that we did not consider the case where the orthogonality assumption is not satisfied because of the omission of a relevant variable; [15] studied the later case and proved that the LS estimates do not converge to their true values). This is proven in Theorem 2.1:

Theorem 2.1. Let 𝑦_𝑡 be generated by (2.4). (i)Let 𝑥_𝑡 be generated by (2.2). The innovations of both DGPs, 𝑢_𝑧,𝑡, for 𝑧=𝑦,𝑥, are independent white noises with zero-mean and constant variance 𝜎²_𝑧; use 𝑦_𝑡 and 𝑥_𝑡 to estimate regression (1.1) by LS. Hence, as 𝑇→∞,(a)𝛼̂→^𝑝𝜇_𝑦, (b)𝛽̂‌→^𝑝𝛽_𝑦, (c)𝑇^−3/2𝑡_𝛽=𝑂_𝑝(1), (d)𝜎̂→^2𝑝𝜎²_𝑦+𝜌²_𝑥𝜎²_𝑥, (e)𝑇²(1−𝑅²)→^𝑝12(𝜎²_𝑦+𝜌²_𝑥𝜎²_𝑥)/(𝛽_𝑦𝜇_𝑥)². (ii)Let 𝑥_𝑡 be generated by (2.3). The innovations of both DGPs, 𝑢_𝑧,𝑡, for 𝑧=𝑦,𝑥, are independent white noises with zero-mean and constant variance 𝜎²_𝑧; use 𝑦_𝑡 and 𝑥_𝑡 to estimate regression (1.1) by LS. Hence, as 𝑇→∞, (a)𝛼̂→^𝑝𝜇_𝑦, (b)𝛽̂‌→^𝑝𝛽_𝑦, (c)𝑇^−3/2𝑡_𝛽=𝑂_𝑝(1), (d)𝜎̂→^2𝑝𝜎²_𝑦+𝜌²_𝑥𝜎²_𝑥, (e)𝑇²(1−𝑅²)→^𝑝12(𝜎²_𝑦+𝜌²_𝑥𝜎²_𝑥)/[𝛽_𝑦(𝜇_𝑥+𝜌_𝑦1)]².

Proof. See Appendix A.

These asymptotic results show that a relationship between the innovations of 𝑦_𝑡 and 𝑥_𝑡—as stated by DGPs (2.2), (2.3), and (2.4)—does not obstruct the consistency of LS estimates when the variables are nonstationary and cointegrated (our results are in line with those of [8]). In other words, the failure of the orthogonality assumption does not preclude adequate asymptotic properties of LS. Furthermore, it can be said that 𝑥_𝑡 is weakly exogenous for the estimation of 𝜇_𝑦 and 𝛽_𝑦 but not for the estimation of 𝜎². The formula of the variance is noteworthy and the asymptotic expression of 𝑡_𝛽 depends on the values of 𝜎²_𝑥, 𝜎²_𝑦, and 𝜌_𝑥.

In order to emphasize the relevance of this result, we modified the DGPs of the variables in an effort to strengthen the link between the DGPs and the literature on simultaneous equations. The modifications are twofold and appear in the following propositions. As in Theorem 2.1, the results in proposition 1 are made under the assumption that innovations are i.i.d processes.

Proposition 2.2. Let 𝑦_𝑡 and 𝑥_𝑡 be generated by 𝑦_𝑡−𝛽_𝑦𝑥_𝑡−𝜇_𝑦=𝑢_𝑦,𝑡,−𝛽_𝑥𝑦_𝑡+𝑥_𝑡−𝜇_𝑥−𝛾_𝑥𝑡=𝑢_𝑥,𝑡,(2.5) where 𝑢_𝑧,𝑡, for 𝑧=𝑥,𝑦, are independent white noises with zero mean and variance 𝜎²_𝑧. Let these variables be used to estimate regression (1.1) by LS. Hence, as 𝑇→∞, (1)𝛼̂→^𝑝𝜇_𝑦, (2)𝛽̂‌→^𝑝𝛽_𝑦, (3)𝑇^−3/2𝑡_𝛽=𝑂_𝑝(1), (4)𝜎̂→^2𝑝𝜎²_𝑦, (5)𝑇²(1−𝑅²)→^𝑝12𝜎²_𝑦(1−𝛽_𝑥𝛽_𝑦)²/(𝛽_𝑦𝛾_𝑥)².

Proof. See Appendix A.

Proposition 2.3. Let 𝑦_𝑡 and 𝑥_𝑡 be generated by 𝑦_𝑡−𝛽_𝑦𝑥_𝑡−𝜇_𝑦=𝑢_𝑦,𝑡,−𝛽_𝑥𝑦_𝑡+𝑥_𝑡−𝑋₀−𝜇_𝑥𝑡=𝜉_𝑥,𝑡,(2.6) where 𝑢_𝑧,𝑡, for 𝑧=𝑥,𝑦, are independent white noises with zero mean and variance 𝜎²_𝑧, and 𝜉_𝑥,𝑡=∑^𝑡_𝑖=0𝑢_𝑥,𝑖. Let these variables be used to estimate regression (1.1) by LS. Hence, as 𝑇→∞, (1)𝛼̂→^𝑝𝜇_𝑦, (2)𝛽̂‌→^𝑝𝛽_𝑦, (3)𝑇^−3/2𝑡_𝛽=𝑂_𝑝(1), (4)𝜎̂→^2𝑝𝜎²_𝑦, (5)𝑇²(1−𝑅²)→^𝑝12𝜎²_𝑦(1−𝛽_𝑥𝛽_𝑦)²/(𝜇_𝑥𝛽_𝑦)².

Proof. See Appendix A.

The two systems, represented in (2.5) and (2.6), bear a striking resemblance to classical examples of simultaneous equations in econometrics. The fundamental variations are, (i) a deterministic trend in the variable 𝑥_𝑡 in system (2.5) and (ii) a stochastic as well as a deterministic trend in system (2.6). The asymptotics of LS estimates do not show significant differences from those in Theorem 2.1. Note, however, that 𝑥_𝑡 is weakly exogenous for the estimation of 𝜇_𝑦, 𝛽_𝑦, and 𝜎²_𝑦. The main result is in fact identical, that is, the failure of orthogonality between 𝑥_𝑡 and the error term does not preclude the estimates from converging to their true values.

Asymptotic properties of LS estimators clearly provide an encouraging perspective in time-series econometrics. Notwithstanding, we should bear in mind that asymptotic properties may be a poor finite-sample approximation. In order to observe the behavior of LS estimates in finite samples, we present two Monte Carlo experiments. Firstly, we represent graphically the convergence process of 𝛽̂‌ towards its true value, 𝛽. In accordance with asymptotic results, 𝛽̂‌−𝛽→^𝑝0 as 𝑇→∞. We reproduce the behavior of the later difference in figure 1. The variables 𝑥 and 𝑦 are generated according to (2.3) and (2.4), respectively. The sample size varies from 50 to 700 whilst 𝛽_𝑦 goes from −5 to 5. The remaining parameters appear below the figure.

A brief glance at Figure 1 reveals that the asymptotic results stated in Theorem 2.1 approximate conveniently the finite-sample results for 𝑇>150. For smaller sample sizes, it can be seen that the difference between the parameter and its estimates corresponds usually to approximately 1.5% or less of the value of the former (we tried different variables in the 𝑦 axis (𝜌_𝑥,𝜌_𝑦1,𝜌_𝑦2,𝜎²_𝑦,𝜎²_𝑦,…); all of these trials produced similar figures).

Figure 1 

Finite-sample behavior of 𝛽̂‌−𝛽𝑦: sample size and 𝛽𝑦 range from 𝑇=50-700 and −5 to 5, respectively; 𝜌𝑦1=−0.15; 𝜌𝑦2=1.5; 𝜌𝑥=4; 𝜎2𝑦=𝜎2𝑥=1; 𝜇𝑦=4.20; 𝜇𝑥=0.70. Number of replications (for each coordinate): 20; grid density: 70×30.

The second Monte Carlo is built upon the same basis. In Table 1, each cell indicates the sample mean of 𝛽̂‌−𝛽_𝑦 and, below, its estimated standard deviation (in parentheses). The number of replications is 10,000. The parameter values used in the simulation are explicit within the table. The variables, 𝑥 and 𝑦, are generated according to (2.3) and (2.4), respectively. Sample size ranges from 𝑇=50-700; 𝜌_𝑦1=−0.15; 𝜌_𝑥=4; 𝜎²_𝑦=𝜎²_𝑥=1; 𝜇_𝑦=4.20; the error term is a white noise with variance 𝜎²_𝜖=1.

Table 1 

Finite-sample behavior of 𝛽̂‌−𝛽_𝑦: mean and standard deviation.

Table 1 shows that LS estimates of a nonstationary relationship with a nonorthogonality problem quickly converge to their true value; with a sample size as small as 50 observations, the difference between 𝛽_𝑦 and its estimate averages, at most, 0.015, and represents a deviation from the true value of 1.5%; in many other cases, the deviation is even smaller, of order 10⁻³–10⁻⁴. These differences tend to diminish further as the sample size grows. In fact, when there are 700 observations, the order of magnitude of such differences oscillates between 10⁻⁵–10⁻⁸. We performed the same experiment with autocorrelated disturbances AR(1) with 𝜙=0.7 (data available upon request); using such disturbances severely deteriorates the efficiency of the LS estimates although 𝛽̂‌−𝛽_𝑦 still converges to zero; we do not focus on this issue because, as mentioned earlier, neither autocorrelation nor heteroskedasticity are under scrutiny in this work.

3. Concluding Remarks

Using cointegrated variables in an LS regression where the regressor is not independent of the error term does not preclude the method from yielding consistent estimates. In other words, it is proven that, under these circumstances, the regressor remains weakly exogenous for the estimation of 𝜇_𝑦 and 𝛽_𝑦 (and for 𝜎²_𝑦 in systems (2.5) and (2.6)) as defined by [12]. Furthermore, the finite-sample evidence indicates that LS provide good estimates even in samples of a practical size.

Notwithstanding, one should note the striking resemblance between the properties of the DGPs used in the propositions and those of variables belonging to a classical simultaneous-equation model. It may be possible that the estimation of such models, even if the macroeconomic variables they are nourished with are not stationary, would yield correct estimates. Of course, such a possibility rules out the existence of structural shifts, parameter instability, omission of a relevant variable, or any other major assumption failure.

Appendix

A. Proof of Theorem 2.1 and Propositions 2.2 and 2.3

The estimated specification in Theorem 2.1 and Propositions 2.2 and 2.3 is 𝑦_𝑡=𝛼+𝛽𝑥_𝑡+𝑢_𝑡. In all three cases, we employ the following classical LS formulae (all sums run from 𝑡=1 to 𝑇 unless otherwise specified): (i)𝐵=(𝑋^𝑋)⁻¹𝑋^𝑌, (ii)𝜎̂²=𝑇⁻¹∑𝑢²_𝑡̂=𝑇⁻¹[∑𝑦²_𝑡+𝛼̂²𝑇+𝛽̂‌²∑𝑥²_𝑡−2𝛼̂∑𝑦_𝑡−2𝛽̂‌∑𝑦_𝑡𝑥_𝑡+2𝛼̂𝛽̂‌∑𝑥_𝑡], (iii)𝑡_𝛽=𝛽̂‌((𝑋^𝑋)⁻¹₂₂𝜎̂²)^−1/2, (iv)𝑅²=1−RSS/TSS,

where 𝑋^𝑋=⎛⎜⎝𝑇𝑥_𝑡𝑥_𝑡𝑥²_𝑡⎞⎟⎠,𝑋^𝑌=⎛⎜⎝𝑦_𝑡𝑦_𝑡𝑥_𝑡⎞⎟⎠,𝐵=𝛼̂𝛽̂‌,(A.1)

SSR=∑𝑢²_𝑡̂, TSS=∑(𝑦_𝑡−𝑦)²=∑𝑦²_𝑡−𝑇⁻¹(∑𝑦_𝑡)², and (𝑋^𝑋)⁻¹₂₂ is the element in row 2, column 2, of the (𝑋^𝑋)⁻¹ matrix.

To obtain the asymptotics of 𝛼̂, 𝛽̂‌, 𝜎̂², 𝑡_𝛽, and 𝑅² we need to ascertain the behaviour of the following expressions when 𝑇→∞: ∑𝑥_𝑡, ∑𝑦_𝑡, ∑𝑥²_𝑡, ∑𝑦²_𝑡, and ∑𝑥_𝑡𝑦_𝑡. The behavior of these expressions varies depending on the DGP of the variables 𝑥_𝑡 and 𝑦_𝑡. We present such behavior for the DGPs underlying Theorem 2.1 and Propositions 2.2 and 2.3. All of the orders in probability stated in the underbraced sums can be found in [5, 13, 16–18]. It is important to clarify that the computation of the asymptotics follows [5] and was assisted by Mathematica; we thus rewrote below the expressions written as Mathematica code.

A.1. Theorem 2.1: First Result

The expressions needed to compute the asymptotic values of 𝛼̂, 𝛽̂‌, 𝜎̂², and 𝑅² are𝑥_𝑡=𝑋₀𝑇+𝜇_𝑥𝑡+𝜉_{𝑥,𝑡−1𝑂𝑝𝑇3/2},𝑥²_𝑡=𝑋²₀𝑇+𝜇²_𝑥𝑡²+𝜉²_{𝑥,𝑡−1𝑂𝑝𝑇2}+2𝑋₀𝜇_𝑥𝑡+2𝑋₀𝜉_{𝑥,𝑡−1}+2𝜇_𝑥𝜉_{𝑥,𝑡−1}𝑡_{𝑂𝑝𝑇5/2},𝑥_𝑡𝑢_𝑥,𝑡=𝑋₀𝑢_{𝑥,𝑡𝑂𝑝𝑇1/2}+𝜇_𝑥𝑢_𝑥,𝑡𝑡_{𝑂𝑝𝑇3/2}+𝜉_{𝑥,𝑡−1}𝑢_{𝑥,𝑡𝑂𝑝(𝑇)},𝑥_𝑡𝑢_𝑦,𝑡=𝑋₀𝑢_𝑦,𝑡+𝜇_𝑥𝑢_𝑦,𝑡𝑡+𝜉_{𝑥,𝑡−1}𝑢_𝑦,𝑡,𝑦_𝑡=𝜇_𝑦𝑇+𝛽_𝑦𝑥_𝑡+𝑢_𝑦,𝑡+𝜌_𝑥𝑢_𝑥,𝑡,𝑦²_𝑡=𝜇²_𝑦𝑇+𝛽²_𝑦𝑥²_𝑡+𝜌²_𝑥𝑢²_𝑥,𝑡+𝑢²_𝑦,𝑡+2𝜇_𝑦𝛽_𝑦𝑥_𝑡+2𝜇_𝑦𝜌_𝑥𝑢_𝑥,𝑡+𝑢_𝑦,𝑡+2𝛽_𝑦𝜌_𝑥𝑥_𝑡𝑢_𝑥,𝑡+𝑥_𝑡𝑢_𝑦,𝑡+2𝜌_𝑥𝑢_𝑥,𝑡𝑢_{𝑦,𝑡𝑂𝑝𝑇1/2},𝑥_𝑡𝑦_𝑡=𝜇_𝑦𝑥_𝑡+𝛽_𝑦𝑥²_𝑡+𝑥_𝑡𝑢_𝑦,𝑡+𝜌_𝑥𝑥_𝑡𝑢_𝑥,𝑡,(A.1) where 𝜉_𝑦,𝑡=∑^𝑡_𝑖=1𝑢_𝑦,𝑖 and 𝑌₀ is an initial condition. The sums including solely the deterministic trend component are𝑡=12𝑇²+𝑇,𝑡²=162𝑇³+3𝑇²+𝑇.(A.2) The code in this case is represented below. To understand it, a brief glossary is required and appears in Table 2.

Table 2 

Glossary of the Mathematica code.

These expressions were written as Mathematica 7.0 code.ClearAll,𝑆_𝑡=12𝑇²+𝑇,𝑆_𝑡2=162𝑇³+3𝑇²+𝑇,𝑆_𝑥=𝑋₀𝑇+𝜇_𝑥𝑆_𝑡+𝑆_𝜉𝑥𝑇^3/2,𝑆_𝑥2=𝑋²₀𝑇+𝜇²_𝑥𝑆_𝑡2+𝑆_𝜉𝑥2𝑇²+2𝑋₀𝜇_𝑥𝑆_𝑡+2𝑋₀𝑆_𝜉𝑥𝑇^3/2+2𝜇_𝑥𝑆_𝜉𝑥𝑡𝑇^5/2,𝑆_𝑦=𝜇_𝑦𝑇+𝛽_𝑦𝑆_𝑥+𝜌_𝑥𝑆_𝑢𝑥𝑇^1/2+𝑆_𝑢𝑦𝑇^1/2,𝑆_𝑥𝑢𝑥=𝑋₀𝑆_𝑢𝑥𝑇^1/2+𝜇_𝑥𝑆_𝑢𝑥𝑡𝑇^3/2+𝑆_{𝜉𝑥𝑢𝑥}𝑇,𝑆_𝑥𝑢𝑦=𝑋₀𝑆_𝑢𝑦𝑇^1/2+𝜇_𝑥𝑆_𝑢𝑦𝑡𝑇^3/2+𝑆_{𝜉𝑥𝑢𝑦}𝑇,𝑆_𝑦2=𝜇²_𝑦𝑇+𝛽²_𝑦𝑆_𝑥2+𝜌²_𝑥𝑆_𝑢𝑥2𝑇+𝑆_𝑢𝑦2𝑇+2𝜇_𝑦𝛽_𝑦𝑆_𝑥+2𝜇_𝑦𝜌_𝑥𝑆_ux𝑇^1/2+𝑆_𝑢𝑦𝑇^1/2+2𝛽_𝑦𝜌_𝑥𝑆_𝑥𝑢𝑥+𝑆_𝑥𝑢𝑦+2𝜌_𝑥𝑆_{𝑢𝑥𝑢𝑦}𝑇^1/2,𝑆_𝑥𝑦=𝜇_𝑦𝑆_𝑥+𝛽_𝑦𝑆_𝑥2+𝜌_𝑥𝑆_𝑥𝑢𝑥+𝑆_𝑥𝑢𝑦.(A.3)

A.2. Theorem 2.1: Second Result

The expressions needed to compute the asymptotic values of 𝛼̂, 𝛽̂‌, 𝜎̂², and 𝑅² appear below. Note that ∑𝑦_𝑡, ∑𝑦²_𝑡, and ∑𝑦_𝑡𝑥_𝑡 are identical to the ones presented in the previous appendix and have been therefore omitted𝑥_𝑡=𝑋₀𝑇+𝜇_𝑥+𝜌_𝑦1𝑡+𝜉_{𝑥,𝑡−1}+𝜌_𝑦2𝜉_{𝑦,𝑡−1},𝑥²_𝑡=𝑋²₀𝑇+𝜇_𝑥+𝜌_𝑦1²𝑡²+𝜉²_{𝑥,𝑡−1}+𝜌²_𝑦2𝜉²_{𝑦,𝑡−1}+2𝑋₀𝜇_𝑥+𝜌_𝑦1𝑡+2𝑋₀𝜉_{𝑥,𝑡−1}+2𝜌_𝑦2𝑋₀𝜉_{𝑦,𝑡−1}+2𝜇_𝑥+𝜌_𝑦1𝜉_{𝑥,𝑡−1}𝑡+2𝜌_𝑦2𝜇_𝑥+𝜌_𝑦1𝜉_{𝑦,𝑡−1}𝑡+2𝜌_𝑦2𝜉_{𝑥,𝑡−1}𝜉_{𝑦,𝑡−1𝑂𝑝𝑇2},𝑥_𝑡𝑢_𝑥,𝑡=𝑋₀𝑢_𝑥,𝑡+𝜇_𝑥+𝜌_𝑦1𝑢_𝑥,𝑡𝑡+𝜉_{𝑥,𝑡−1}𝑢_𝑥,𝑡+𝜌_𝑦2𝜉_{𝑦,𝑡−1}𝑢_𝑥,𝑡,𝑥_𝑡𝑢_𝑦,𝑡=𝑋₀𝑢_𝑦,𝑡+𝜇_𝑥+𝜌_𝑦1𝑢_𝑦,𝑡𝑡+𝜉_{𝑥,𝑡−1}𝑢_𝑦,𝑡+𝜌_𝑦2𝜉_{𝑦,𝑡−1}𝑢_𝑦,𝑡.(A.1) The code in this case is represented below. ClearAll,𝑆_𝑡=12𝑇²+𝑇,𝑆_𝑡2=162𝑇³+3𝑇²+𝑇,𝑆_𝑥=𝑋₀𝑇+𝜇_𝑥+𝜌_𝑦1𝑆_𝑡+𝑆_𝜉𝑥𝑇^3/2+𝜌_𝑦2𝑆_𝜉𝑦𝑇^3/2,𝑆_𝑥2=𝑋²₀𝑇+𝜇_𝑥+𝜌_𝑦1²𝑆_𝑡2+𝑆_𝜉𝑥2𝑇²+𝜌²_𝑦2𝑆_𝜉𝑦2𝑇²+2𝑋₀𝜇_𝑥+𝜌_𝑦1𝑆_𝑡+2𝑋₀𝑆_𝜉𝑥𝑇^3/2+2𝑋₀𝜌_𝑦2𝑆_𝜉𝑦𝑇^3/2+2𝜇_𝑥+𝜌_𝑦1𝑆_𝜉𝑥𝑡𝑇^5/2+2𝜇_𝑥+𝜌_𝑦1𝜌_𝑦2𝑆_𝜉𝑦𝑡𝑇^5/2+2𝜌_𝑦2𝑆_{𝜉𝑦𝜉𝑥}𝑇²,𝑆_𝑦=𝜇_𝑦𝑇+𝛽_𝑦𝑆_𝑥+𝜌_𝑥𝑆_𝑢𝑥𝑇^1/2+𝑆_𝑢𝑦𝑇^1/2,𝑆_𝑥𝑢𝑥=𝑋₀𝑆_𝑢𝑥𝑇^1/2+𝜇_𝑥+𝜌_𝑦1𝑆_𝑢𝑥𝑡𝑇^3/2+𝑆_{𝜉𝑥𝑢𝑥}𝑇+𝜌_𝑦2𝑆_{𝜉𝑦𝑢𝑥}𝑇,𝑆_𝑥𝑢𝑦=𝑋₀𝑆_𝑢𝑦𝑇^1/2+𝜇_𝑥+𝜌_𝑦1𝑆_𝑢𝑦𝑡𝑇^3/2+𝑆_{𝜉𝑥𝑢𝑦}𝑇+𝜌_𝑦2𝑆_{𝜉𝑦𝑢𝑦}𝑇,𝑆_𝑦2=𝜇²_𝑦𝑇+𝛽²_𝑦𝑆_𝑥2+𝜌²_𝑥𝑆_𝑢𝑥2𝑇+𝑆_𝑢𝑦2𝑇+2𝜇_𝑦𝛽_𝑦𝑆_𝑥+2𝜇_𝑦𝜌_𝑥𝑆_𝑢𝑥𝑇^1/2+𝑆_𝑢𝑦𝑇^1/2+2𝛽_𝑦𝜌_𝑥𝑆_𝑥𝑢𝑥+𝑆_𝑥𝑢𝑦+2𝜌_𝑥𝑆_{𝑢𝑥𝑢𝑦}𝑇^1/2,𝑆_𝑥𝑦=𝜇_𝑦𝑆_𝑥+𝛽_𝑦𝑆_𝑥2+𝜌_𝑥𝑆_𝑥𝑢𝑥+𝑆_𝑥𝑢𝑦.(A.2)  

A.3. Proposition 2.2

First note that DGP (2.5) can be written as𝑦_𝑡=𝜇_𝑦+𝛽_𝑦𝑥_𝑡+𝑢_𝑦,𝑡,𝑥_𝑡=𝐶₁+𝐶₂𝑡+𝛽_𝑥𝐶₀𝑢_𝑦,𝑡+𝑢_𝑥,𝑡𝐶₀,(A.1) where, 𝐶₀=1−𝛽_𝑥𝛽_𝑦, 𝐶₁=(𝜇_𝑥+𝜇_𝑦𝛽_𝑥)/𝐶₀, and 𝐶₂=𝛾_𝑥/𝐶₀. The expressions needed to compute the asymptotic values of 𝛼̂, 𝛽̂‌, 𝜎̂², and 𝑅² are𝑥_𝑡=𝐶₁𝑇+𝐶₂𝑡+𝛽_𝑥𝐶₀𝑢_𝑦,𝑡+1𝐶₀𝑢_𝑥,𝑡,𝑥²_𝑡=𝐶²₁𝑇+𝐶²₂𝑡²+𝛽_𝑥𝐶₀²𝑢²_𝑦,𝑡+1𝐶₀²𝑢²_𝑥,𝑡+2𝐶₁𝐶₂𝑡+2𝐶₁𝛽_𝑥𝐶₀𝑢_𝑦,𝑡+2𝐶₁𝐶₀𝑢_𝑥,𝑡+2𝐶₂𝛽_𝑥𝐶₀𝑢_𝑦,𝑡𝑡+2𝐶₂𝐶₀𝑢_𝑥,𝑡𝑡+2𝛽_𝑥𝐶²₀𝑢_𝑥,𝑡𝑢_𝑦,𝑡,𝑥_𝑡𝑢_𝑦,𝑡=𝐶₁𝑢_𝑦,𝑡+𝐶₂𝑢_𝑦,𝑡𝑡+𝛽_𝑥𝐶₀𝑢²_𝑦,𝑡+1𝐶₀𝑢_𝑥,𝑡𝑢_𝑦,𝑡,𝑦_𝑡=𝜇_𝑦𝑇+𝛽_𝑦𝑥_𝑡+𝑢_𝑦,𝑡,𝑦²_𝑡=𝜇²_𝑦𝑇+𝛽²_𝑦𝑥²_𝑡𝑢²_𝑦,𝑡+2𝜇_𝑦𝛽_𝑦𝑥_𝑡+2𝜇_𝑦𝑢_𝑦,𝑡+2𝛽_𝑦𝑥_𝑡𝑢_𝑦,𝑡,𝑥_𝑡𝑦_𝑡=𝜇_𝑦𝑥_𝑡+𝛽_𝑦𝑥²_𝑡+𝑥_𝑡𝑢_𝑦,𝑡.(A.2) The code in this case is represented below.ClearAll,𝑆_𝑡=12𝑇²+𝑇,𝑆_𝑡2=162𝑇³+3𝑇²+𝑇,𝐶₀=1−𝛽_𝑥𝛽_𝑦,𝐶₁=𝜇_𝑥+𝜇_𝑦𝛽_𝑥1−𝛽_𝑥𝛽_𝑦,𝐶₂=𝛾_𝑥1−𝛽_𝑥𝛽_𝑦,𝑆_𝑥=𝐶₁𝑇+𝐶₂𝑆_𝑡+𝛽_𝑥𝐶₀𝑆_𝑢𝑦𝑇^1/2+1𝐶₀𝑆_𝑢𝑥𝑇^1/2,𝑆_𝑥2=𝐶₁²𝑇+𝐶₂²𝑆_𝑡2+𝛽_𝑥𝐶₀²𝑆_𝑢𝑦2𝑇+1𝐶₀²𝑆_𝑢𝑥2𝑇+2𝐶₁𝐶₂𝑆_𝑡+2𝐶₁𝛽_𝑥𝐶₀𝑆_𝑢𝑦𝑇^1/2+2𝐶₁𝐶₀𝑆_𝑢𝑥𝑇^1/2+2𝐶₂𝛽_𝑥𝐶₀𝑆_𝑢𝑦𝑡𝑇^3/2+2𝐶₂𝐶₀𝑆_𝑢𝑥𝑡𝑇^3/2+2𝛽_𝑥𝐶₀²𝑆_{𝑢𝑥𝑢𝑦}𝑇^1/2,𝑆_𝑦=𝜇_𝑦𝑇+𝛽_𝑦𝑆_𝑥+𝑆_𝑢𝑦𝑇^1/2,𝑆_𝑥𝑢𝑦=𝐶₁𝑆_𝑢𝑦𝑇^1/2+𝐶₂𝑆_𝑢𝑦𝑡𝑇^3/2+𝛽_𝑥𝐶₀𝑆_𝑢𝑦2𝑇+1𝐶₀𝑆_{𝑢𝑥𝑢𝑦}𝑇^1/2,𝑆_𝑦2=𝜇_𝑦²𝑇+𝛽_𝑦²𝑆_𝑥2+𝑆_𝑢𝑦2𝑇+2𝜇_𝑦𝛽_𝑦𝑆_𝑥+2𝜇_𝑦𝑆_𝑢𝑦𝑇^1/2+2𝛽_𝑦𝑆_𝑥𝑢𝑦,𝑆_𝑥𝑦=𝜇_𝑦𝑆_𝑥+𝛽_𝑦𝑆_𝑥2+𝑆_𝑥𝑢𝑦.(A.3)

A.4. Proposition 2.3

As in the previous appendix, first rewrite DGP (2.5) as𝑦_𝑡=𝜇_𝑦+𝛽_𝑦𝑥_𝑡+𝑢_𝑦,𝑡,𝑥_𝑡=𝐷₁+𝐷₂𝑡+𝛽_𝑥𝐷₀𝑢_𝑦,𝑡+𝜉_{𝑥,𝑡−1}𝐷₀,(A.1) where, 𝐷₀=1−𝛽_𝑥𝛽_𝑦, 𝐷₁=(𝑋₀+𝜇_𝑦𝛽_𝑥)/𝐷₀, and 𝐷₂=𝜇_𝑥/𝐷₀. The expressions needed to compute the asymptotic values of 𝛼̂, 𝛽̂‌, 𝜎̂², and 𝑅² appear below. Note that ∑𝑦_𝑡, ∑𝑦²_𝑡, and ∑𝑦_𝑡𝑥_𝑡 are identical to the ones presented in the previous appendix and have been therefore omitted𝑥_𝑡=𝐷₁𝑇+𝐷₂𝑡+𝛽_𝑥𝐷₀𝑢_𝑦,𝑡+1𝐷₀𝜉_{𝑥,𝑡−1},𝑥²_𝑡=𝐷²₁𝑇+𝐷²₂𝑡²+𝛽_𝑥𝐷₀²𝑢²_𝑦,𝑡+1𝐷₀²𝜉²_{𝑥,𝑡−1}+2𝐷₁𝐷₂𝑡+2𝐷₁𝛽_𝑥𝐷₀𝑢_𝑦,𝑡+2𝐷₁𝐷₀𝜉_{𝑥,𝑡−1}+2𝐷₂𝛽_𝑥𝐷₀𝑢_{𝑦,𝑡−1}𝑡+2𝐷₂𝐷₀𝜉_{𝑥,𝑡−1}𝑡+2𝛽_𝑥𝐷²₀𝜉_{𝑥,𝑡−1}𝑢_𝑦,𝑡,𝑥_𝑡𝑢_𝑦,𝑡=𝐷₁𝑢_𝑦,𝑡+𝐷₂𝑢_𝑦,𝑡𝑡+𝛽_𝑥𝐷₀𝑢²_𝑦,𝑡+1𝐷₀𝜉_{𝑥,𝑡−1}𝑢_𝑦,𝑡.(A.2) The code in this case is represented below:ClearAll,𝑆_𝑡=12𝑇²+𝑇,𝑆_𝑡2=162𝑇³+3𝑇²+𝑇,𝐷₀=1−𝛽_𝑥𝛽_𝑦,𝐷₁=𝑋₀+𝜇_𝑦𝛽_𝑥1−𝛽_𝑥𝛽_𝑦,𝐷₂=𝜇_𝑥1−𝛽_𝑥𝛽_𝑦,𝑆_𝑥=𝐷₁𝑇+𝐷₂𝑆_𝑡+𝛽_𝑥𝐷₀𝑆_𝑢𝑦𝑇^1/2+1𝐷₀𝑆_𝜉𝑥𝑇^3/2,𝑆_𝑥2=𝐷₁²𝑇+𝐷₂²𝑆_𝑡2+𝛽_𝑥𝐷₀²𝑆_𝑢𝑦2𝑇+1𝐷₀²𝑆_𝜉𝑥2𝑇²+2𝐷₁𝐷₂𝑆_𝑡+2𝐷₁𝛽_𝑥𝐷₀𝑆_𝑢𝑦𝑇^1/2+2𝐷₁𝐷₀𝑆_𝜉𝑥𝑇^3/2+2𝐷₂𝛽_𝑥𝐷₀𝑆_𝑢𝑦𝑡𝑇^3/2+2𝐷₂𝐷₀𝑆_𝜉𝑥𝑡𝑇^5/2+2𝛽_𝑥𝐷₀²𝑆_{𝜉𝑥𝑢𝑦}𝑇;𝑆_𝑦=𝜇_𝑦𝑇+𝛽_𝑦𝑆_𝑥+𝑆_𝑢𝑦𝑇^1/2,𝑆_𝑥𝑢𝑦=𝐷₁𝑆_𝑢𝑦𝑇^1/2+𝐷₂𝑆_𝑢𝑦𝑡𝑇^3/2+𝛽_𝑥𝐷₀𝑆_𝑢𝑦2𝑇+1𝐷₀𝑆_{𝜉𝑥𝑢𝑦}𝑇,𝑆_𝑦2=𝜇_𝑦²𝑇+𝛽_𝑦²𝑆_𝑥2+𝑆_𝑢𝑦2𝑇+2𝜇_𝑦𝛽_𝑦𝑆_𝑥+2𝜇_𝑦𝑆_𝑢𝑦𝑇^1/2+2𝛽_𝑦𝑆_𝑥𝑢𝑦,𝑆_𝑥𝑦=𝜇_𝑦𝑆_𝑥+𝛽_𝑦𝑆_𝑥2+𝑆_𝑥𝑢𝑦.(A.3)

A.5. Computation of the Asymptotics

The previous three appendices provide the Mathematica code of ∑𝑥_𝑡, ∑𝑦_𝑡, ∑𝑥²_𝑡, ∑𝑦²_𝑡, and ∑𝑥_𝑡𝑦_𝑡 for different DGP combinations. We now present the code that computes the asymptotics of (1.1) in any such combination. Note that the code computes the asymptotics in the following order: the matrix (𝑋^𝑋)⁻¹, (𝑋^𝑋)⁻¹₂₂, 𝛼̂, 𝛽̂‌, 𝜎̂², 𝑡_𝛽, and 1−𝑅². Comments appear inside parentheses (*---*).

(^∗Matrix(𝑋^𝑋)^∗)

Mx=^{𝑇𝑆𝑥𝑆}_𝑥^𝑆_𝑥2;

(^∗InverseofMatrix(𝑋^𝑋)^∗)𝑖Mx=Inverse[Mx];

(^∗Element1,1of(𝑋^𝑋)^−1∗)R1=Extract[𝑖Mx,{1,1}];(^∗Element1,2of(𝑋^𝑋)^−1∗)R2=Extract[𝑖Mx,{1,2}];(^∗Element2,1of(𝑋^𝑋)^−1∗)R3=Extract[𝑖Mx,{2,1}];(^∗Element2,2of(𝑋^𝑋)^−1∗)R4=Extract[𝑖Mx,{2,2}];

(^∗Factorization^∗)R40=Factor[R4];(^∗Numerator^∗)R4num=Numerator[R40];(^∗Denominator^∗)R4den=Denominator[R40];

(^∗Highestpowerof𝑇innumerator^∗)

K1=Exponent[R4num,𝑇],

(^∗Highestpowerof𝑇indenominator^∗)

K2=Exponent[R4den,𝑇];(^∗Limitofthenumeratordividedby𝑇^K1∗)

R4num2=Limit[Expand[R4num/𝑇^K1],T→∞];

(^∗Limitofthedenominatordividedby𝑇^K2∗)

R4den2=Limit[Expand[R4den/𝑇^K2],𝑇→∞];(^∗LimitoftheElement2,2of(𝑋^𝑋)⁻¹multipliedby𝑇^k1/𝑇^K2∗)R42=Factor[Expand[(R4num2/R4den2)∗𝑇^K1/𝑇^K2]];

(^∗Parameter𝛼̂^∗)

P10=Factor[Expand[R1𝑆_𝑦+R2𝑆_𝑥𝑦]];P11num=Numerator[P10];K5=Exponent[P11num,𝑇];Anum=Limit[Expand[P11num/𝑇^𝐾5],𝑇→∞];P12den=Denominator[P10];K6=Exponent[P12den,𝑇];Aden=Limit[Expand[P12den/𝑇^K6],𝑇→∞];Apar=Factor[Expand[(Anum/Aden)∗𝑇^K5/𝑇^K6]]

(^∗Parameter𝛽̂‌^∗)

P20=Factor[Expand[R3𝑆_𝑦+R4𝑆_𝑥𝑦]];P21num=Numerator[P20];K7=Exponent[P21num,𝑇];Bnum=Limit[Expand[P21num/𝑇^K7],𝑇→∞];P22den=Denominator[P20];K8=Exponent[P22den,𝑇];Bden=Limit[Expand[P22den/𝑇^K8],𝑇→∞];Bpar=Factor[Expand[(Bnum/Bden)∗𝑇^K7/𝑇^K8]]

(^∗Parameter𝜎̂^2∗)

P40=Factor[Expand[𝑆_𝑦2+P10²𝑇+P20²𝑆_𝑥2−2P10𝑆_𝑦−2P20𝑆_𝑥𝑦+2P10P20𝑆_𝑥]];P41num=Numerator[P40];K11=Exponent[P41num,𝑇];Vnum=Factor[Limit[Expand[P41num/T^K11],𝑇→∞]];P42den=Denominator[P40];K12=Exponent[P42den,𝑇];Vden=Factor[Limit[Expand[P42den/𝑇^K12],𝑇→∞]];Vpar=Factor[Expand[𝑇⁻¹∗(Vnum/Vden)∗𝑇^K11/𝑇^K12]]

(^∗SSR/TSS^∗)

P50=Factor[Expand[P40/(𝑆_𝑦2−𝑇(𝑆_𝑦/𝑇)²)]];P51num=Numerator[P50];K13=Exponent[P51num,𝑇];Rcnum=Factor[Limit[Expand[P51num/𝑇^K13],𝑇→∞]];P52den=Denominator[P50];K14=Exponent[P52den,𝑇];Rcden=Factor[Limit[Expand[P52den/𝑇^K14],𝑇→∞]];Rc=Factor[Expand[(Rcnum/Rcden)∗𝑇^K13/𝑇^K14]];

(^∗𝑡_𝛽^∗)

𝑡_𝛽=FullSimplify[Bpar/(Vpar∗R42)^−1/2]

(^∗1−𝑅^2∗)

P70=FullSimplify[Rc].

Acknowledgments

The authors would like to thank an anonymous referee for insightful comments. The opinions in this paper correspond to the authors and do not necessarily reflect the point of view of Banco de México.