LONG RUN MONEY DEMAND IN SOUTH AFRICA: A COINTEGRATED VAR APPROACH

Transcription

1 LONG RUN MONEY DEMAND IN SOUTH AFRICA: A COINTEGRATED VAR APPROACH Abdurohman Ali Hussien MSc. Economics, University of Copenhagen, Denmark. Shakeel Ahmed, MSc. Economics, University of Copenhagen, Denmark. Muhammad Yousaf, PhD student of Economics, University of Tartu, Estonia Abstract The argument for controlling inflation through monetary targeting assumes the presence of a stable money demand relation. Though monetary targeting is no more the major monetary policy instrument in South Africa, money demand relation is still one of the channels through which monetary policy affects inflation, which makes is it reasonable to see if a stable long run money demand relation exists in South Africa. Using a Cointegrated VAR method, this study attempts to see if a stable long run money demand relation exists in South Africa. The VAR model constitutes short term interest rate, long term interest rate, inflation, exchange rate, real money and real income as system variables. The findings suggest three cointegrated relations: a relation between long run constant money velocity and inflation; a relation between exchange rate and the short term interest rate, with the shift dummy included; and a relation between the bond rate, inflation and real income. The empirical evidence, however, does not find a stable long run money demand relation unlike previous studies. Neither does the empirical finding support the hypothesis that inflation responds to deviation from a stable long run money demand relation. Key Words: Money Demand, VAR model, South Africa, Cointegration, Monetary Policy. Classification: Journal article 1. Introduction Optimal stabilization policy is believed to bring about a potential welfare gain to citizens. Different countries including South Africa use monetary policy along with fiscal policy to achieve this objective. The presence of a stable long run money demand is at the center of the argument for controlling inflation through monetary targeting. Like many other central banks, the South African Reserve Bank (hereafter SARB) adopted a monetary policy based on pre-announced broad money (M3) target from 1986 until 2000, when the monetary policy regime began to be inflation targeting. Though monetary targeting is no more the major monetary policy instrument in South Africa, money demand relation is still one of the channels through which monetary policy affect inflation, which makes it reasonable to see whether a stable long run money demand relation exists in South Africa. Few studies were made in the area with different methodology for different time series data. Wesso( 2002) and Todani (2005), for instance, have both found a stable demand for money function in the South African economy. The increasing openness of the South African economy since the 1990s may have affected the money demand relation since then. This paper therefore tries to see if the stable money demand relation exist using a Cointegration VAR method including the exchange rate variable in the information set for the period

2 2. The Model and Data 2.1. The Theory Model As is commonly the case, the long run money demand relation is specified as follows: M = Y β 1 (R b R s ) β 2 ( P R s ), for β 1, β 2 > 0 (1) M in equation (1) above is the Real money demand and is correlated positively with real income, Y; and short term interest rate, Rs; and negatively with the long term bond rate, Rb and the inflation rate, P. The bond rate and the inflation rate as a return to alternative assets and goods respectively measure the opportunity cost of holding money. The short term interest rate is considered as the own rate of return to money. The change in exchange rate variable is included in the model as a measure of opportunity cost of holding money. People may prefer to hold US dollar than the rand as a result of excess liquidity that induce them to expect future inflation that lower the value of rand. For instance, excess liquidity may induce people to expect future inflation that lower the value of the rand. As a result, they prefer to hold the US dollar to the rand. The theory model suggests two stationary relations: a constant money velocity (M-Y) relation and a relation between inflation, the short term interest tare and the long term bond rate (Rb-Rs and P-Rs). The money demand equation specified above can be obtained as a linear combination of the two cointegration relations. Whether the empirical evidence supports the theoretical hypothesis is to be seen later in the analysis. The model consists of six system variables: the short term interest rate(rs), the long term bond rate (Rb), the inflation rate ( P), the change in the exchange rate (DifE), real money (M) and real GDP (Y) where each variable is defined as follows: M=ln M3-lnCPI, M3 being the broad monetary aggregate and CPI is the consumer price index Rs=90 day rate /1200 Rb= 10 year bond rate/1200 Y= lngdp DifE= ln (exchange rate of South African rand against the dollar) P= lncpi As can be seen from the graphs in Fig 1, extraordinary events happened in 1998:06 on Rs, Rb, P and E; and in 2001 on E. The former event corresponds to lifting the exchange rate target policy that resulted in depreciation of rand against the dollar by as high as 25.5% (see Aron and Muellbauer, 2006). Through raising cost of import this event caused an inflation hike of 6% which in turn cause a significant rise in Rs and Rb. The second event corresponds to the exchange rate shock in 2001 when the rand depreciated by 42% against the dollar. This event was largely unexpected (see Bhundia and Gottaschalk 2003). The latter is treated as a shift dummy while the former was found to have only a transitory impact. The graphs for real money and real income in the levels have a linear trend that is restricted to the co-integration relations as one could 98

3 Abdourohman Ali Hussien, Shakeel Ahmed, Muhammad Yousaf, not expect a priory that the trend cancels in the co- integration relation before formally testing if it could be excluded. Fig.1. The graphs for the system variables (in levels and differences) s hort term i nteres t rate Di fex c hange rate bond rate 2. 6 real m oney i nfl ati on real i nc om e D s hort term i nteres t rate D Dex c hange rate D bond rate D real m oney D i nfl ati on D real i nc o m e 99

4 2.2. The statistical model A priori a VAR (2) model can be specified as follows: X t =Γ 1 ΔX t-1 + ΠX t-1 + ΦD t + μ + ε t,ε t IN (0,Ω),t = 1,,T (2) Where X t = [R st, R bt, Δ P t, E, M, Y], D t = [D s 0112] D s refers to the shift dummy μ is a constant. The trend and the shift dummy are restricted to the cointegration relation while the constant is unrestricted to indicate a linear trend in the levels Data Monthly observations for short term interest rate, exchange rate, broad money, consumer price index are collected from the Ecowin database. Quarterly data for real GDP is interpolated using monthly data for mining production from the same database. Monthly observation for long term bond rate is found in the IMF database. The variables are transformed as described in section Estimation and Discussion of the Results 3.1. Estimation of the Unrestricted VAR (2) model The unrestricted VAR (2) model is estimated and the results together with the misspecification tests are reported in Appendix A.1. The misspecification tests suggest that the model is not very well specified. In the preferred model (conditional on the exclusion of the trend and weak exogenity of Rs), however, the specification improved enough for inference to be made. LM tests of order one and two show that the null hypothesis of no autocorrelation is accepted. Some ARCH effects seem to be present. Inference to moderate ARCH effects, however, could be taken as robust (see Rahbek et al, 2002) The preferred model The test for the exclusion of the trend term is accepted for rank two as shown in Appendix A2.1. As can be seen from the univariate statistics and the graphs for the residuals, much of the non-normality problem seem to come from Rs that induced for test weak exogenity for Rs. The model improves conditional on weak exogenity of Rs. For both rank of two or three weak exogenity of Rs is accepted (see appendix A2.2). The statistical model shown by equation (2) above is modified in that the trend term is excluded from the cointegration relation and Rs is weakly exogenous. 100

5 3.3. Lag length determination As can be seen from appendix A.2, H-Q and SC tests suggest a lag length of two and one respectively. Second order no-autocorrelation for VAR (2) is accepted. The lag reduction tests, however, suggest a lag length of 5 as opposed to the above tests. It also suggests a choice of VAR (2) against VAR (3) while at the same time suggesting that VAR (4) and VAR (5) against VAR (2). As it is argued in Juselius(2006), this may be due to the time varying seasonality in the data that is not captured by the seasonal dummies.that leads to a commonly chosen VAR model of lag Rank Determination The standard asymptotic table for trace test, the trace test with bartlet correction and the simulated tables that account for the restricted shift dummy suggest a rank of three. For further evidence of rank choice, the roots of the companion matrix for different choice of rank are reported in Appendix A.4. The 3 rd column in the α matrix in Appendix A.1 significantly adjusts to the third cointegration relation for all variables except the inflation rate The largest roots left over with rank two and three based on the roots of the companion matrix are and respectively and it is not clear whether the later is different from a unit root. Most of the evidences however point to a rank choice of three Hypothesis Testing Appendix A5.1 shows that M and Y can be excluded from the co-integration relation for both ranks of two and three. The interest is, however, on constant long run money velocity, M-Y and not on M and Y individually. A test for exclusion of long run constant money velocity (M-Y) is shown to be accepted in Appendix A5.5. However I keep them in the cointegration relation as they are needed for the test of long run money demand stability. In fact it is shown in appendix 6 that a cointegration relation between M-Y and inflation was found. Test for the unit vector in α shows that for rank of two, inflation and exchange rate have only transitory effect on the other variables while the remaining have permanent impact(see Appendix A5.3). Weak exogenity of constant long run money velocity M-Y is accepted, confirming that M-Y is not a stationary relation The identified structure From the beta vectors and the results of the hypothesis tests, testing the following three cointegration relations is suggested. First, the relation between constant long run money demand and inflation; second, the relation between the bond rate, inflation and real income; third, the relation between exchange rate, the short term interest rate and real income. The identified structure is estimated as shown in Appendix A.6. The graphs for the cointegration relations misspecification test and the recursive constancy tests for the identified structure are also reported. The evidence also does not support the hypothesis that money stock adjusts to a long run money demand relation. It is shown in appendix A6.1 that the real money equation has a positive 101

6 coefficient on the 3rd α vector which is against the theoretical hypothesis that a stable long run money demand exists. The negative coefficient for the inflation equation on the 3rd α vector together with the unstable money demand relation is not consistent with the view that inflation is affected by a deviation from a stable long run money demand relation. The increasing openness of the South African economy since the 1990s may have caused a change in the long run money demand relation. The unstable money demand relation as evidenced by this empirical finding is considered as a reason for abandoningmonetary aggregate targeting by increasing number of central banks including SARB and adopting inflation targeting instead (see also Todani,2005). From the PI matrix in appendix A.6, one can see that money stock is not adjusting. Exchange rate depreciates for increase in liquidity but not significantly so. Exchange rate appreciation is also observed for an increase in short term interest rate. This may be due to the increasing openness of the South African economy that led to the inflow of capital for a rise in the short term interest rate. Real money demand decrease for a rise in the bond rate confirming the theory that money and bond are substitutes. Though insignificant, a positive relation between real money and short term interest rate was found confirming the theoretical expectation. An IS type relation, however insignificant, was found between bond rate and real income. Real income is seen to increase as a result of depreciating rand which seems to be due to an increase in competitiveness. Considering the C-matrix below, the following points can be summarized: The fact that money stock is permanently driven by its own shock confirms the previous finding that it is not adjusting. The bond rate has a significant negative impact on real income (seems an IS effect) Empirical shocks to real money and real income seems to be the major pushing forces of the economy. The Long-Run Impact Matrix, C RB DLP DIFE M Y RB (1.181) (1.279) (0.382) (2.857) (2.515) DLP (0.340) (0.686) (0.299) (4.146) (2.551) DIFE (-1.521) (-1.495) (-0.401) (-2.080)(2.361) M (-1.549) (-0.885) (-0.029) (4.551) (-1.262) Y (-2.276) (-1.754) (-0.312) (1.945) (0.753) 102

7 4. Conclusion This paper attempts to find out whether a stable longrun money demand relation exists in the South African economy using a cointegration VAR approach. Short term interest rate, long term interest rate, inflation, exchange rate, real money and real income are included as system variables. The findings suggest three long run stationary relations: a relation between long run constant money velocity and inflation; a relation between exchange rate and the short term interest rate, with the shift dummy included; and a relation between the bond rate, inflation and real income. Also, an IS type relation between the bond rate and real income was found. The empirical evidence, however, doesnot find a stable long run money demand relation unlike previous studies. Money stock was not found to adjust to a long run money demand relation. Neither does the empirical finding support the hypothesis that inflation responds to deviation from a stable long run money demand relation. The finding is in line with the argument that increasing volatility in broad money that caused an increasing number of central banks including SARB to abandon monetary targeting as a monetary policy instrument and adopting inflation targeting instead. 5. References Aron, J. and Muellbauer,J. (2006) Review of monetary policy in south Africa since 1994, Journal of African Economies, Vol. 16 (5), July, PP Juselius, K. (2006), The cointegrated VAR model: Methodology and applications, oxford univerisyt press. Nell, K.S. (2003), The stability of M3 money demand and monetary growth targets: The case of South Africa, Journal of Development Studies, Vol. 39, February, PP Todani, K.R. (2005), A cointegrated VAR model of M3 demand in south Africa, South African Reserve Bank, Working Paper. Wesso, G.R. (2002), Broad money demand and financial liberalization in south Africa, South African Reserve Bank, Working Paper. 103

8 6. Appendices Appendix A.1: Estimation of the baseline VAR(2) and misspecification tests BETA(transposed) RS RB DLP DIFE M Y C(2001:12) TREND Beta(1) Beta(2) Beta(3) Beta(4) Beta(5) Beta(6) ALPHA Alpha(1) Alpha(2) Alpha(3) Alpha(4) Alpha(5) Alpha(6) DRS (0.693) (-1.569) (4.936) (-0.186) (0.139) (-1.574) DRB (0.164) (-2.304) (3.464) (2.946) (-0.878) (-0.854) DDLP (4.972) (6.513) (0.162) (0.646) (0.001) (-1.655) DDIFE ( ) (3.384) (2.033) (1.435) (-0.834) (-0.321) DM (0.946) (0.519) (3.191) (0.498) (2.228) (1.516) DY (1.079) (1.098) (2.226) (-1.774) (-2.451) (1.222) 104

9 PI RB DLP DIFE M Y C(2001:12) TREND DRS (-2.861) (-2.590) (2.380) (1.030) (2.124) (-0.922) (-0.688) (-3.996) DRB (0.581) (-4.577) (2.699) (1.000) (2.815) (-2.263) (-2.547) (-2.243) DDLP (-0.230) (2.342) (-7.578) (3.599) (0.471) (1.098) (1.791) (-1.921) DDIFE (-1.116) (-2.932) (0.686) ( ) (1.592) (-0.515) (-7.328) (-2.304) DM (-0.788) (-1.927) (-0.018) (0.775) (-0.861) (1.141) (1.199) (-2.074) DY (-2.532) (0.983) (-0.786) (0.561) (2.577) (-2.543) (1.713) (0.221) Log-Likelihood =

10 RESIDUAL ANALYSIS Tests for Autocorrelation Ljung-Box(40): ChiSqr(1368) = [0.000] LM(1): ChiSqr(36) = [0.048] LM(2): ChiSqr(36) = [0.096] Test for Normality: ChiSqr(12) = [0.000] Test for ARCH: LM(1): ChiSqr(441) = [0.000] LM(2): ChiSqr(882) = [0.000] Univariate Statistics Mean Std.Dev Skewness Kurtosis Maximum Minimum DRS DRB DDLP DDIFE DM DY ARCH(2) Normality R-Squared DRS [0.000] [0.000] DRB [0.000] [0.000] DDLP [0.857] [0.751] DDIFE [0.407] [0.001] DM [0.478] [0.651] DY [0.003] [0.017]

11 Graphs for the residuals DY A c tual and Fi tted Autoc orrel ati ons Lag S tandardiz ed R es idual s H i stogram SB-DH: Chi Sqr(2) = 8.20 [0.02] K-S = 0.99 [5% C.V. = 0.07] J -B: Chi Sqr(2) = 7.92 [0.02] DM A c tual and Fi tted Autoc orrel ati ons Lag S tandardiz ed R es idual s H i stogram SB-DH: Chi Sqr(2) = 0.86 [0.65] K-S = 0.96 [5% C.V. = 0.07] J -B: Chi Sqr(2) = 0.50 [0.78] D D IF E A c tual and Fi tted Autoc orrel ati ons Lag 5. 0 S tandardiz ed R es idual s H i stogram SB-DH: Chi Sqr(2) = [0.00] K-S = 0.90 [5% C.V. = 0.07] J -B: Chi Sqr(2) = [0.00] D DL P Ac tual and Fitted Autoc orrel ati ons Lag 3 2 S tandardiz ed R es idual s H i stogram SB-DH: Chi Sqr(2) = 0.57 [0.75] K-S = 0.91 [5% C.V. = 0.07] J -B: Chi Sqr(2) = 0.20 [0.91]

12 D RB Ac tual and Fitted Autoc orrel ati ons Lag S tandardiz ed R es idual s H i stogram SB-DH: Chi Sqr(2) = [0.00] K-S = 0.86 [5% C.V. = 0.07] J -B: Chi Sqr(2) = [0.00] D RS Ac tual and Fitted Autoc orrel ati ons Lag 6 4 Standardi z ed Res iduals H i stogram SB-DH: Chi Sqr(2) = [0.00] K-S = 0.91 [5% C.V. = 0.07] J -B: Chi Sqr(2) = [0.00]

13 Appendix A.2: The preferred model A2.1. TEST OF EXCLUSION r DGF 5% C.V. RS RB DLP DE M Y C(2001:12) TREND [0.423] [0.087] [0.045] [0.000] [0.815] [0.741] [0.000] [0.643] [ [0.438] 0.010] [0.000] [0.000] [0.969] [0.273] [0.000] [0.436] [0.046] [0.002] [0.000] [0.000] [0.258] [0.284] [0.000] [0.008] Notes: LR-test, Chi-Square(r), P-values in brackets. A2.2. TEST OF WEAK EXOGENEITY r DGF 5% C.V. RB DLP DIFE M Y [0.705] [0.002] [0.000] [0.380] [0.325] [0.351] [0.000] [0.000] [0.627] [0.353] [0.486] [0.000] [0.000] [0.540] [0.445] [0.131] [0.000] [0.000] [0.557] [0.170] Notes: LR-Test, Chi-Square(r), P-values in brackets. 109

14 RESIDUAL ANALYSIS of the preffered model Tests for Autocorrelation Ljung-Box(40): ChiSqr(950) = [0.000] LM(1): ChiSqr(25) = [0.152] LM(2): ChiSqr(25) = [0.169] Test for Normality: ChiSqr(10) = [0.016] Test for ARCH: LM(1): ChiSqr(225) = [0.005] LM(2): ChiSqr(450) = [0.001] Univariate Statistics Mean Std.Dev Skewness Kurtosis Maximum Minimum DRB DDLP DDIFE DM DY ARCH(2) Normality R-Squared DRB [0.023] [0.989] DDLP [0.367] [0.724] DDIFE [0.581] [0.019] DM [0.665] [0.540] DY [0.004] [0.014]

15 Appendix A.3: Lag length determination MODEL SUMMARY Model k T Regr Log-Lik SC H-Q LM(1) LM(k) VAR(5) VAR(4) VAR(3) VAR(2) VAR(1) Lag Reduction Tests: VAR(4) << VAR(5) : ChiSqr(30) = [0.001] VAR(3) << VAR(5) : ChiSqr(60) = [0.002] VAR(3) << VAR(4) : ChiSqr(30) = [0.207] VAR(2) << VAR(5) : ChiSqr(90) = [0.001] VAR(2) << VAR(4) : ChiSqr(60) = [0.046] VAR(2) << VAR(3) : ChiSqr(30) = [0.053] VAR(1) << VAR(5) : ChiSqr(120) = [0.000] VAR(1) << VAR(4) : ChiSqr(90) = [0.000] VAR(1) << VAR(3) : ChiSqr(60) = [0.000] VAR(1) << VAR(2) : ChiSqr(30) = [0.000] 111

16 Appendix A4 Rank determination I(1)-ANALYSIS p-r r Eig.Value Trace Trace* Frac95 P-Value P-Value* WARNING: Critical/P-values correspond to a model with level shifts. WARNING: The Bartlett Corrections correspond to the 'Basic Model'. SIMULATION OF THE ASYMPTOTIC TRACE TEST DISTRIBUTION I(1)-ANALYSIS p-r r Eig.Value Trace Trace* Frac95 P-Value P-Value* WARNING: The Bartlett Corrections correspond to the 'Basic Model.' 112

17 The Roots of the COMPANION MATRIX // Model: H(0) Real Imaginary Modulus Argument Root Root Root Root Root Root Root The Roots of the COMPANION MATRIX // Model: H(1) Real Imaginary Modulus Argument Root Root Root Root Root Root Root The Roots of the COMPANION MATRIX // Model: H(2) Real Imaginary Modulus Argument Root Root Root Root Root Root Root

18 The Roots of the COMPANION MATRIX // Model: H(3) Real Imaginary Modulus Argument Root Root Root Root Root Appendix A.5: Hypothesis testing A.5.1. TEST OF EXCLUSION r DGF 5% C.V. RB DLP DIFE M Y RS C(2001:12) [0.055] [0.014] [0.000] [0.817] [0.974] [0.452] [0.000] [0.011] [0.000] [0.000] [0.971] [0.660] [0.512] [0.000] [0.021] [0.000] [0.000] [0.714] [0.543] [0.718] [0.000] [0.006] [0.000] [0.000] [0.783] [0.488] [0.182] [0.000] Notes: LR-test, Chi-Square(r), P-values in brackets. 114

19 A.5.2: TEST OF STATIONARITY LR-test, Chi-Square(5-r), P-values in brackets. r DGF 5% C.V. RB DLP DIFE M Y [0.000] [0.000] [0.019] [0.000] [0.000] [0.000] [0.015] [0.155] [0.000] [0.000] [0.506] [0.492] [0.544] [0.377] [0.474] [0.909] [0.236] [0.731] [0.290] [0.337] A.5.3 TEST OF UNIT VECTOR IN ALPHA LR-test, Chi-Square(5-r), P-values in brackets. r DGF 5% C.V. RB DLP DIFE M Y [0.000] [0.000] [0.019] [0.000] [0.000] [0.000] [0.282] [0.905] [0.000] [0.000] [0.504] [0.530] [0.945] [0.114] [0.312] [0.466] [0.812] [0.917] [0.037] [0.177] 115

20 A.5.4. TEST OF WEAK EXOGENEITY LR-Test, Chi-Square(r), P-values in brackets. r DGF 5% C.V. RB DLP DIFE M Y [0.705] [0.002] [0.000] [0.380] [0.325] [0.351] [0.000] [0.000] [0.627] [0.353] [0.486] [0.000] [0.000] [0.540] [0.445] [0.131] [0.000] [0.000] [0.557] [0.170] A.5.5. Exclusion of M-Y TEST OF RESTRICTED MODEL: CHISQR(3) = [0.616] BARTLETT CORRECTION: CHISQR(3) = [0.741] (Correction Factor: 1.437) *** NOTE: The correction factor is based on the 'Basic Model'. T DLP DIFE M Y RS C(2001:12) Beta(1) Beta(2) Beta(3)

21 A5.6. Weak exogenity of M-Y TEST OF RESTRICTED MODEL: CHISQR(3) = [0.978] BETA(transposed) DLP DIFE M Y RS C(2001:12) Beta(1) Beta(2) Beta(3) Appendix A.6: The identified structure A6.1. TEST OF RESTRICTED MODEL: CHISQR(8) = [0.379] THE MATRICES BASED ON 3 COINTEGRATING VECTORS: BETA(transposed) DLP DIFE M Y RS C(2001:12) Beta(1) (.NA) (.NA) (.NA) (.NA) (.NA) (3.49) (6.33) Beta(2) (.NA) (.NA) (.NA) (.NA) (15.56) (.NA) (.NA) Beta(3) (NA) ( ) (.NA) (.NA) (.NA) (.NA) (.NA) 117

22 ALPHA Alpha(1) Alpha(2) Alpha(3) DRB (0.592) (-1.857) (1.543) DDLP (3.526) (4.815) (-1.950) DDIFE ( ) (-0.783) (0.883) DM (0.916) (-2.778) (3.230) DY (0.269) (-1.234) (1.737) PI DLP DIFE M Y RS C(2001:12) DRB (-1.857) (1.512) (0.592) (1.543) (-1.815) (0.592) (0.592) DDLP (4.815) (-8.979) (3.526) (-1.950) (4.324) (3.526) (3.526) DDIFE (-0.783) (0.066) ( ) (0.883) (-0.809) ( ) ( ) DM (-2.778) (-0.009) (0.916) (3.230) (-2.888) (0.916) (0.916) DY (-1.234) (-0.753) (0.269) (1.737) (-1.339) (0.269) (0.269) 118

23 A6.2. Graphs for the cointegrated relations 0.20 Beta1'*Z1(t) Beta1'*R1(t) Beta2'*Z1(t) Beta2'*R1(t) Beta3'*Z1(t) Beta3'*R1(t)

24 A.6.3. Residual analysis for the identified structure Tests for Autocorrelation Ljung-Box(40): ChiSqr(960) = [0.000] LM(1): ChiSqr(25) = [0.172] LM(2): ChiSqr(25) = [0.239] Test for Normality: ChiSqr(10) = [0.003] Test for ARCH: LM(1): ChiSqr(225) = [0.007] LM(2): ChiSqr(450) = [0.001] Univariate Statistics Mean Std.Dev Skewness Kurtosis Maximum Minimum DRB DDLP DDIFE DM DY ARCH(2) Normality R-Squared DRB [0.062] [0.302] DDLP [0.370] [0.755] DDIFE [0.782] [0.004] DM [0.785] [0.274] DY [0.000] [0.107]