Objective 3

Chapter 3 Research Methodology

3.3 Objective 3

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𝑗 is the optimal lag length determined by AIC, SIC and HQC; we use the number of lags that minimize the criteria;

𝑒_𝑖𝑡−1 is the error correction term and it represents how far the variables are from the equilibrium relationship. The error-correction mechanism estimates how, in the event of an imbalance variables adjust towards parity so as to preserve the long-run relationship. If the set of estimated coefficients (ʎ_2𝑖 to ʎ_4𝑖) on lagged independent variables are non-zero, then there is short-run causality. If the ECM coefficient ʎ_1𝑖 is negative and significant then there is long-run causality.

The same procedure is conducted involving net remittance volatility (𝑓𝑟𝜎) in place of net foreign portfolio investment volatility (𝑓𝑝𝑖𝜎).

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the capital flow volatility study by Broto et al. (2011). Thirdly, the variable inflation as proxied by the CPI estimates fluctuations in the general price level of a basket of household commodities as supported in the SVAR study by Raghavan and Silvapulle (2008). In addition, the study utilized domestic interest rates proxied by bank rate (IN) that is widely supported in the literature (Ngalawa and Viegi, 2011). This is the rate at which the central bank lends short-term funds to local financial institutions. In this study, the global interest rate is employed and is proxied by the FFR as supported in Kutu and Ngalawa (2016) and Neumann et al. (2009). This is an exogenous variable that is defined as the overnight lending rate by depository institutions in the US and is included to capture the open economies of the SADC countries.

3.3.3 MODEL SPECIFICATION

Stock and Watson (2001) define VAR as an econometric model in which each variable is explained by its own lagged values, taking into account the current and past values of the remaining variables.

According to Gujarati (2003), all the variables are assumed to be endogenous. VAR models are dynamic and make use of economic theory. They are important tools for explaining the dynamic behavior of economic and financial data (Lütkepohl, 2012). Unrestricted VAR with unit roots is undesirable in that it is inconsistent and can mislead policy analysis. Accordingly, this study employs a six-variable panel VAR methodology to estimate the model using the impulse response function and variance decomposition in levels.

Following Cheng (2006), suppose the low-income SADC countries can be represented by the following equation:

𝐴𝑌_𝑖𝑡 = 𝛼_𝑖𝑡+ 𝐵₁𝑌_𝑖𝑡−1+ 𝐵₂𝑌_𝑖𝑡−2+. . … . . +𝐵_𝑝𝑌_{𝑖𝑡−𝑝}+ 𝜆_𝑖𝑋_𝑖𝑡+ 𝐷𝜀_𝑖𝑡 ` 3.6 where 𝐴 is an invertible square (6 × 6) matrix describing the contemporaneous relationships among the variables; 𝑌_𝑖𝑡 is a (6 × 1) vector of endogenous variables such that (𝑌_𝑖𝑡 = 𝑌_1𝑖𝑡, 𝑌_2𝑖𝑡,… … 𝑌_𝑛𝑖𝑡); 𝛼_𝑖𝑡 is a (6 × 1) vector of constants; 𝐵_𝑖 is a (6 × 6) matrix of coefficients of lagged endogenous variables (𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑖 = 1 … … … … . . 𝑝); 𝜆_𝑖 and 𝑋_𝑖𝑡 are the coefficients and vectors of the exogenous variables, respectively, capturing external shocks; 𝐷 is a (6 × 6) matrix whose non-zero off-diagonal components accommodate direct impacts of some shocks on more than one endogenous variable in the system; and 𝜀_𝑖𝑡 is a vector of uncorrelated or orthogonal white- noise structural disturbances.

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The P-VAR shown in equation (3.6) cannot be estimated directly due to the contemporaneous feedback inherent in the VAR process (Enders, 2004). The system incorporates feedback because the endogenous variables are allowed to influence each other in the current and past realisation time path of 𝐴𝑌_𝑖𝑡. The parameters in the system are unidentifiable and their values cannot be determined because their coefficients are not known. According to Ngalawa and Viegi (2011), the information in the system can be recovered by estimating a reduced form 𝑉𝐴𝑅 implicit in the equations. Pre-multiplying equation (3.6) by an inverse of 𝐴 gives:

𝑌_𝑖𝑡 = 𝐴⁻¹_𝛼𝑖𝑡+ 𝐴⁻¹𝐵₁𝑌_𝑖𝑡−1+ 𝐴⁻¹𝐵₂𝑌_𝑖𝑡−2+ … +𝐴⁻¹𝐵_𝑝𝑌_{𝑖𝑡−𝑝}+ 𝐴⁻¹𝜆_𝑖𝑋_𝑖𝑡+ 𝐴⁻¹𝐷𝜀_𝑖𝑡 3.7 One can denote 𝐴⁻¹_𝛼𝑖𝑡 = 𝐶, 𝐴⁻¹𝐵₁… . . 𝐴⁻¹𝐵_𝑝 = 𝐴_𝑖 𝑓𝑜𝑟 𝑖 = 1 … 𝑝, 𝐴⁻¹𝜆_𝑖 = 𝛼𝑖 𝑎𝑛𝑑 𝐴⁻¹𝐷𝜀_𝑡 = 𝜇_𝑖𝑡. Hence, equation (3.7) becomes:

𝑌_𝑖𝑡 = 𝐶 + 𝐴₁𝑌_𝑖𝑡−1+ 𝐴₂𝑌_𝑡−2+ … + 𝐴_𝑝𝑌_𝑡−𝑝 + 𝛼_𝑖𝑋_𝑡 + _𝑖𝑡 3.8 The first equation (3.6) is called a long form 𝑉𝐴𝑅 or primitive VAR system where all variables have contemporaneous effects on each other, while equation (3.8) is called a reduced form 𝑉𝐴𝑅 or a 𝑉𝐴𝑅 in standard form in which all the right-hand side variables are predetermined at time t and no variable has a direct contemporaneous effect on another in the model. In addition, the error term (𝜇_𝑡) is a composite of shocks in 𝑌_𝑡 (Enders, 2004). By substitution, the reduced form of equation (3.8) can be further stated as follows:

𝑌_𝑖𝑡 = 𝛽𝑖 + 𝐴(𝐿)𝑌_𝑖𝑡+ 𝐵(𝐿)𝑋_𝑖𝑡+ 𝜇_𝑖𝑡 ………..…(3.9) where 𝑌_𝑖𝑡 and 𝑋_𝑖𝑡 are 6 𝑋 1 vectors of variables given by

𝑌_𝑖𝑡 = (𝑁𝐹𝑃𝐼𝜎, 𝑅𝐺𝐷𝑃, 𝑀2, 𝐼𝑁, 𝐶𝑃𝐼,)………..…...(3.10) 𝑋_𝑖𝑡 = (𝐹𝐹𝑅)…..………...…..…….…(3.11) The first model that captures the dynamic effects of net foreign portfolio investment (NFPI) volatility is represented by equation 3.10 while equation 3.12 below is the second model that captures the dynamic impacts of net foreign remittance (NFR) volatility in low-income SADC economies:

𝑌_𝑖𝑡 = (𝑁𝐹𝑅𝜎, 𝑅𝐺𝐷𝑃, 𝑀2, 𝐼𝑁, 𝐶𝑃𝐼,)………..……….……...(3.12) 𝑋_𝑖𝑡 = (𝐹𝐹𝑅)…..………..……...(3.13)

115 Where:

NFPI represents net foreign portfolio investment volatility NFR represents net foreign remittance volatility

RGDP represents the real GDP growth rate M2 represents money supply

IN represents the domestic interest rate CPI represents the consumer price index

FFR represent the federal funds rate that captures the external variable

𝑌_𝑖𝑡 is a (6 × 1) vector of endogenous variables such that 𝑌_𝑖𝑡 = 𝑌_1𝑡, 𝑌_2𝑡,… 𝑌_6𝑡. 𝛽_𝑖 is a (6 × 1) vector of constants representing country specific intercept terms.

𝐴(𝐿) and 𝐵(𝐿) are matrices of polynomial lags that capture the relationship between the endogenous variables and their lags.

𝜇_𝑖𝑡 is a vector of reduced random disturbances (error term).

Therefore, we can summarize the above set of links between innovations and structural shocks as given by Cheng (2006:12) in the following matrices as:

[ 𝜀_𝑖𝑡^𝐹𝐹𝑅 𝜀_𝑖𝑡^{𝑅𝐺𝐷𝑃} 𝜀_𝑖𝑡^NFPIσ

𝜀_𝑖𝑡^MS 𝜀_𝑖𝑡^𝐶𝑃𝐼 𝜀_𝑖𝑡^𝐼𝑁𝑇 ]

=

[

1 0 0 0 0 0

𝑓₂₁^𝑗 1 0 0 0 0 𝑓₃₁^𝑗 𝑓₃₂^𝑗 1 0 0 0 𝑓₄₁^𝑗 𝑓₄₂^𝑗 𝑓₄₃^𝑗 1 0 0 𝑓₅₁^𝑗 𝑓₅₂^𝑗 𝑓₅₃^𝑗 𝑓₅₄^𝑗 1 0 𝑓₄₁^𝑗 𝑓₄₁^𝑗 𝑓₄₁^𝑗 𝑓₄₁^𝑗 𝑓₄₁^𝑗 1][

𝜇_𝑖𝑡^𝐹𝐹𝑅 𝜇_𝑖𝑡^{𝑅𝐺𝐷𝑃} 𝜇_𝑖𝑡^NFPIσ

𝜇_𝑖𝑡^MS 𝜇_𝑖𝑡^𝐶𝑃𝐼 𝜇_𝑖𝑡^𝐼𝑁𝑇 ]

………(3.14)

116 [

𝜀_𝑖𝑡^𝐹𝐹𝑅 𝜀_𝑖𝑡^{𝑅𝐺𝐷𝑃} 𝜀_𝑖𝑡^NFRσ 𝜀_𝑖𝑡^MS 𝜀_𝑖𝑡^𝐶𝑃𝐼 𝜀_𝑖𝑡^𝐼𝑁𝑇 ]

=

[

1 0 0 0 0 0

𝑓₂₁^𝑗 1 0 0 0 0 𝑓₃₁^𝑗 𝑓₃₂^𝑗 1 0 0 0 𝑓₄₁^𝑗 𝑓₄₂^𝑗 𝑓₄₃^𝑗 1 0 0 𝑓₅₁^𝑗 𝑓₅₂^𝑗 𝑓₅₃^𝑗 𝑓₅₄^𝑗 1 0 𝑓₄₁^𝑗 𝑓₄₁^𝑗 𝑓₄₁^𝑗 𝑓₄₁^𝑗 𝑓₄₁^𝑗 1][

𝜇_𝑖𝑡^𝐹𝐹𝑅 𝜇_𝑖𝑡^{𝑅𝐺𝐷𝑃} 𝜇_𝑖𝑡^NFRσ 𝜇_𝑖𝑡^MS 𝜇_𝑖𝑡^𝐶𝑃𝐼 𝜇_𝑖𝑡^𝐼𝑁𝑇 ]

………(3.15)

Equation 3.14 shows the processes to determine the impact of changes in net foreign portfolio investment volatility in low-income economies while equation 3.15 allows for determination of the impact of changes in net foreign remittances in low-income economies.

The first row estimates the external pressures on the domestic economy from global interest rates represented by the FFR. Transmission of global shocks to the domestic market can be rapid and shocks are transmitted from the global market to the domestic economy and not vice versa (Berkelmans, 2005). For example, in the first equation, global interest rates are assumed to be causing changes in GDP and driving fluctuations in the variability of foreign portfolio and foreign remittance flows as shown in equations 3.14 and 3.15. However, global interest rates are also assumed to be driving their own changes while 𝐺𝐷𝑃 and volatility changes depend on global interest rates and their own changes. It is crucial to realize that some variables in the representation such as 𝐶𝑃𝐼, 𝑀2 𝑎𝑛𝑑 𝐼𝑁 are policy variables under the management of the monetary authorities.

Shocks to these variables are normally subjected to information delays or lags caused by policy makers (Sims and Zha, 2006, Berkelmans, 2005). The way variables influence each other depends on their position in the identification scheme and their ordering is based on theoretical expectations where interest rates is ordered last in the matrices in order to control for inflation in the economy.

3.3.4 JUSTIFICATION FOR P-VAR

The P-VAR approach accounts for endogenous relationships and can summarize empirical relationships without placing too many restrictions on the data used (Berkelmans, 2005). VAR is more powerful than ordinary OLS, GARCH, EGARCH and other approaches in that it makes use of some minimal restrictions known as “Cholesky-one standard deviation” and can accommodate up to seven variables without running out of degrees of freedom (Raghavan and Silvapulle, 2008).

The P-VAR model also enables the determination of unbiased impulse response functions (IRFs)

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as it takes full advantage of the information contained in the cross-sectional dimension of the sample (Góes, 2016). More importantly, it shows the dynamic behavior and response of every variable to a disturbance that occurs in the economy (Van Aarle, Garretsen, and Gobbin, 2003).

In addition, P-VAR in levels allows time and history to determine whether the impact of a shock is temporary or permanent. All the variables are employed with their natural logarithms to minimize heteroscedasticity (Ramaswamy and Sloek, 1998).

3.3.5 LAG LENGTH SELECTION

An ideal lag length ensures that residuals do not suffer from autocorrelation, non-normality and heteroskedacity problems (Hatemi-J and S. Hacker, 2009). The approaches adopted to select an appropriate lag length to use in each equation are: Akaike Information criteria (AIC), Schwarz Information criteria (SIC) and Hannan and Quinn criteria (HQC). We choose the number of lags that minimizes the criteria.

3.3.6 IMPULSE RESPONSE FUNCTION (IRF) AND VARIANCE DECOMPOSITION Like the standard VAR model, P-VAR models provide convenient instruments in the form of impulse-response function and variance decomposition which provide more information on the impact and transmission of shocks and policy innovations. Impulse response functions are vital for analyzing the dynamic interactions between variables in a VAR model and show the effects of shocks on the adjustment path of the variable. IRF also indicates the response of variables to shocks affecting the system and is shown by way of graphs. In contrast, variance decomposition is a measure of the contribution of each type of shock to the forecast error variance and comes in absolute figures or percentages which must add up to 100.

The study used EViews 10 for all the data analyses.