*3.1. Time Series Properties and Estimators*

The long-run relationship between variables in levels can be evaluated using the Engle–Granger two-step residual based procedure (Engle and Granger 1987) and Johansen's system-based reduced rank regression approach (Johansen 1988; Johansen and Juselius 1990). A disadvantage of the Engle and Granger (1987) approach is the small sample bias arising from the exclusion of short-run dynamics (Alam and Quazi 2003). The procedures developed by Johansen (1988) and Phillips and Hansen (1990) require that the variables involved should be *<sup>I</sup>*(1). However, the ARDL bounds testing procedure developed by Pesaran et al. (2001) applies whether the regressors are *<sup>I</sup>*(0), purely *I*(1) or integrated of di fferent orders. Hence, this study adopts the ARDL bounds testing procedure to examine REER misalignment and capital flight from Botswana (Equations (1), (8) and (9)). The bounds testing procedure is advantageous since it can be executed even if the explanatory variables are endogenous. The approach is also suitable for smaller samples. To perform the bounds test, the following steps are followed. First, we determine the stationarity of the variables. Then, the optimal lag for the model is evaluated using the Schwarz–Bayesian information criterion (SBIC). The long-run relationship between the variables in levels is evaluated using the bounds test. Following Pesaran et al. (2001), the error correction models for this study are represented as:

$$\begin{aligned} \Delta LNREER\_{l} &= \beta\_{0} \quad + \sum\_{i=1}^{n1} \beta\_{1i} \Delta LINREER\_{l-i} + \sum\_{i=0}^{n2} \beta\_{2i} \Delta LINOT\_{l-i} + \sum\_{i=0}^{n3} \beta\_{3i} \Delta LINGOV\_{l-i} \\ &+ \sum\_{i=0}^{n4} \beta\_{4i} \Delta GDP\_{l-i} + \sum\_{i=0}^{n5} \beta\_{5i} \Delta FDI\_{l-i} + \sum\_{i=0}^{n6} \beta\_{6i} \Delta AID\_{l-i} \\ &+ \sum\_{i=0}^{n7} \beta\_{7i} \Delta LINPENNES\_{l-i} + \sum\_{i=0}^{n8} \beta\_{8i} \Delta LINEBT\_{l-i} \\ &+ \sum\_{i=0}^{n9} \beta\_{9i} \Delta LINCAPTA\_{l-i} + \delta\_{0} \Delta NREER\_{l-1} + \delta\_{1} LNTOT\_{l-1} \\ &+ \delta\_{2} LNGOV\_{l-1} + \delta\_{3} GDP\_{l-1} + \delta\_{4} FDl\_{l-1} + \delta\_{5} AD\_{l-1} \\ &+ \delta\_{6} LNOEENNES\_{l-1} + \delta\_{7} LNDEBT\_{l-1} + \delta\_{8} LNCAPATAL\_{l-1} + \mu\_{1} \end{aligned} \tag{10}$$

$$\begin{aligned} \Delta \mathcal{K} F\_t &= \beta\_0 + \sum\_{i=1}^{n1} \quad \beta\_{1i} \Delta \mathcal{K} F\_{t-i} + \sum\_{i=1}^{n2} \beta\_{2i} \Delta \mathcal{R} E \mathcal{S} E \mathcal{R} V ES\_{t-i} + \sum\_{i=0}^{n3} \beta\_{3i} \Delta \mathcal{G} D P\_{t-i} \\ &+ \sum\_{i=0}^{n4} \beta\_{4i} \Delta \mathcal{M} F\_{t-i} + \sum\_{i=0}^{n5} \beta\_{5i} \Delta \mathcal{I} R D\_{t-i} + \sum\_{i=0}^{n6} \beta\_{6i} \Delta \mathcal{A} A D\_{t-i} \\ &+ \sum\_{i=0}^{n7} \beta\_{7i} \Delta \mathcal{L} N O P \mathcal{E} N \mathcal{S} S\_{t-i} + \sum\_{i=0}^{n8} \beta\_{8i} \Delta \mathcal{L} N \mathcal{D} B T\_{t-i} \\ &+ \sum\_{i=0}^{n9} \beta\_{9i} \Delta \mathcal{O} V \mathcal{E} R\_{t-i} + \delta\_{0} \mathcal{K} F\_{t-1} + \delta\_{1} \mathcal{R} E \mathcal{S} \mathcal{R} V \mathcal{S} S\_{t-1} + \delta\_{2} \mathcal{G} D P\_{t-1} \\ &+ \delta\_{3} N \mathcal{F}\_{t-1} + \delta\_{4} \mathcal{R} D\_{t-1} + \delta\_{5} \mathcal{A} D\_{t-1} + \delta\_{6} \mathcal{L} N O P \mathcal{E} N \mathcal{S} S\_{t-1} \\ &+ \delta\_{7} \mathcal{L} N \mathcal{D} E \mathcal{B}\_{t-1} + \delta\_{8} \delta V \mathcal{E} R\_{t-1} + \mu\_{t} \end{aligned} \tag{11}$$

$$\begin{aligned} \Delta KF\_{t} &= \beta\_{0} + \sum\_{i=1}^{n1} \quad \beta\_{1i} \Delta KF\_{t-i} + \sum\_{i=1}^{n2} \beta\_{2i} \Delta RESRVES\_{t-i} + \sum\_{i=0}^{n3} \beta\_{3i} \Delta GDP\_{t-i} \\ &\quad + \sum\_{i=0}^{n4} \beta\_{4i} \Delta INF\_{t-i} + \sum\_{i=0}^{n5} \beta\_{5i} \Delta IDD\_{t-i} + \sum\_{i=0}^{n6} \beta\_{6i} \Delta ADID\_{t-i} \\ &\quad + \sum\_{i=0}^{n7} \beta\_{7i} \Delta LNOPEINES\_{t-i} + \sum\_{i=0}^{n8} \beta\_{8i} \Delta LNDEBT\_{t-i} \\ &\quad + \sum\_{i=0}^{n9} \beta\_{9i} \Delta LINDER\_{t-i} + \delta\_{0} \delta F\_{t-1} + \delta\_{1} RESERVES\_{t-1} + \delta\_{2} GDP\_{t-1} \\ &\quad + \delta\_{3} \text{INF}\_{t-1} + \delta\_{4} IRD\_{t-1} + \delta\_{5} \Delta IDD\_{t-1} + \delta\_{6} LNOPEINENES\_{t-1} \\ &\quad + \delta\_{7} \text{LND}DEBT\_{t-1} + \delta\_{8} \delta LINDER\_{t-1} + \mu \end{aligned} \tag{12}$$

The definition of terms is as follows. Δ is the first difference operator. β0 is the regression constant while μ*t* represents the white noise error term. δ0 − δ7 indicate the long-run coefficients while β1 − β8 represent the error correction short-run dynamics. The error-correction models for the above equations after affirming the long-run relationship between the variables is specified as follows:

Δ*LNREERt* = β0 + *n*1 *i*=1 β1*i*Δ*LNREERt*−*<sup>i</sup>* + *n*2 *i*=0 β2*i*Δ*LNTOTt*−*<sup>i</sup>* + *n*3 *i*=0 β3*i*Δ*LNGOVt*−*<sup>i</sup>* + *n*4 *i*=0 β4*i*Δ*GDPt*−*<sup>i</sup>* + *n*5 *i*=0 β5*i*Δ*FDIt*−*<sup>i</sup>* + *n*6 *i*=0 β6*i*Δ*AIDt*−*<sup>i</sup>* + *n*7 *i*=0 β7*i*Δ*LNOPENNESSt*−*<sup>i</sup>* + *n*8 *i*=0 β8*i*Δ*LNDEBTt*−*<sup>i</sup>* + *n*9 *i*=0 β9*i*Δ*LNCAPITALt*−*<sup>i</sup>* + θ*ecmt*−<sup>1</sup> + μ*t* (13) Δ*KFt* = β0 + *n*1 *i*=1 β1*i*Δ*KFt*−*<sup>i</sup>* + *n*2 *i*=1 β2*i*Δ*RESERVESt*−*<sup>i</sup>* + *n*3 *i*=0 β3*i*Δ*GDPt*−*<sup>i</sup>* + *n*4 *i*=0 β4*i*Δ*INFt*−*<sup>i</sup>* + *n*5 *i*=0 β5*i*Δ*IRDt*−*<sup>i</sup>* + *n*6 *i*=0 β6*i*Δ*AIDt*−*<sup>i</sup>* + *n*7 *i*=0 β7*i*Δ*LNOPENNESSt*−*<sup>i</sup>* + *n*8 *i*=0 β8*i*Δ*LNDEBTt*−*<sup>i</sup>* + *n*9 *i*=0 β9*i*Δ*OVERt*−*<sup>i</sup>* + θ*ecmt*−<sup>1</sup> + μ*t* (14) Δ*KFt* = β0 + *n*1 *i*=1 β1*i*Δ*KFt*−*<sup>i</sup>* + *n*2 *i*=1 β2*i*Δ*RESERVESt*−*<sup>i</sup>* + *n*3 *i*=0 β3*i*Δ*GDPt*−*<sup>i</sup>* + *n*4 *i*=0 β4*i*Δ*INFt*−*<sup>i</sup>* + *n*5 *i*=0 β5*i*Δ*IRDt*−*<sup>i</sup>* + *n*6 *i*=0 β6*i*Δ*AIDt*−*<sup>i</sup>* + *n*7 *i*=0 β7*i*Δ*LNOPENNESSt*−*<sup>i</sup>* + *n*8 *i*=0 β8*i*Δ*LNDEBTt*−*<sup>i</sup>* + *n*9 *i*=0 β9*i*Δ*UNDERt*−*<sup>i</sup>* + θ*ecmt*−<sup>1</sup> + μ*t* (15)

where the term *ecmt*−1 is the error-correction term and θ is its coefficient. The term *ecmt*−1 represents the speed of convergence to the equilibrium level if there is a disturbance in the system.

This study further uses the Toda and Yamamoto (1995) approach to Granger causality to investigate the causal relationship among the variables. The ordinary Granger causality test is not suitable for this investigation since the variables under investigation are not all *<sup>I</sup>*(1). Further, Wolde-Rufael (2005) argue that the *F*-statistic in Granger causality may be invalid because the test lacks a standard distribution in cases where the times series is cointegrated. An advantage of the Toda and Yamamoto (1995) approach is its applicability whether a series is *<sup>I</sup>*(0), *<sup>I</sup>*(1), *<sup>I</sup>*(2), non-cointegrated or cointegrated of different arbitrary orders. The test also uses a modified Wald test and a standard autoregressive model in levels form to reduce the likelihood of mistakenly identifying the order of the series (Mavrotas and Kelly 2001). For estimation purposes, the vector autoregression (VAR) system for *MISREER* and the causes of misalignment is represented as:

$$MIISREER\_{l} = a\_{0} + \sum\_{i=1}^{k} a\_{1i}MISRER\_{t-i} + \sum\_{j=k+1}^{d\_{\text{max}}} a\_{2j}MISRER\_{t-j} + \sum\_{i=1}^{k} \phi\_{1i}X\_{t-i} + \sum\_{j=k+1}^{d\_{\text{max}}} \phi\_{2j}X\_{t-j} + \lambda\_{it} \tag{16}$$

$$X\_{l} = \alpha\_{0} + \sum\_{i=1}^{k} \beta\_{1i} X\_{t-i} + \sum\_{j=k+1}^{d\_{\text{max}}} \beta\_{2j} X\_{t-j} + \sum\_{i=1}^{k} \delta\_{1i} MISREER\_{t-i} + \sum\_{j=k+1}^{d\_{\text{max}}} \alpha\_{2j} MISREER\_{t-j} + \lambda\_{2l} \tag{17}$$

where *Xt* is any of the potential causes of misalignment (*LNCAC*, *LNDEBTC*, *EMS* and *LNRGDPC*). The VAR system represents the bivariate relationship between *MISREER* and each potential cause. The causal links examined are as follows: *MISREER* − *LNCAC*; *MISREER* − *LNDEBT*; *MISREER* − *EMS* and *MISREER* − *LNGDPC*. The highest order of integration is denoted by *dmax*. Likewise, the VAR system for *KF* and its causes, *Yt* is represented as:

$$\text{KF}\_{I} = a\wp + \sum\_{i=1}^{k} \alpha\_{1i} \text{KF}\_{I-i} + \sum\_{j=k+1}^{d\_{\text{max}}} \alpha\_{2j} \text{KF}\_{I-j} + \sum\_{i=1}^{k} \phi\_{1i} \text{Y}\_{I-i} + \sum\_{j=k+1}^{d\_{\text{max}}} \phi\_{2j} \text{Y}\_{I-j} + \lambda\_{il} \tag{18}$$

$$Y\_t = \alpha\_0 + \sum\_{i=1}^k \beta\_{1i} Y\_{t-i} + \sum\_{j=k+1}^{d\_{\text{max}}} \beta\_{2j} Y\_{t-j} + \sum\_{i=1}^k \delta\_{1i} K F\_{t-i} + \sum\_{j=k+1}^{d\_{\text{max}}} \alpha\_{2j} K F\_{t-j} + \lambda\_{2t} \tag{19}$$

where *Yt* is any of the determinants of KF. The VAR system represents the bivariate relationship between *KF* and each potential cause. The bivariate causal links examined are as follows:

$$KF-RESER VS; KF-GDP; KF-INF; KF-IRD; KF-AID; KF-LNOP; KF-RNOP; KF-RNOP; KF-RNOP.}$$
