RDP 9510: Modelling Inflation in Australia Appendix 2: Design of the Empirical ECM
November 1995
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This appendix discusses several of the steps taken to obtain the ECM in (17) and examines in greater detail the recursive estimates of the coefficients in (17). This simplification to (17) has numerous motivations. Because of its parsimony and more orthogonal regressors, (17) is more easily interpreted than the unrestricted ADL or VAR. Also, the coefficients in (17) are more precisely estimated than those in the ADL and the VAR, providing tighter inferences generally and higher potential power for tests of misspecification.
Initially, the vector autoregression for the system cointegration analysis is simplified from a fourthorder VAR to a firstorder VAR, where the variables in the VARs are p, ulc, ip, pet, a constant, and centered seasonal dummies. Table A1 reports the F statistics and related calculations for that simplification, where the longest lag on all variables is deleted repeatedly from the VAR. None of the F statistics comparing the initial, intermediate, and final VARs is significant; and the Schwarz criterion becomes steadily more negative as the lag length is shortened. The Schwarz criterion in effect adjusts a measure of the model's goodness of fit (the log of the determinant of the estimated error variance matrix) for the model's degree of parsimony. A smaller Schwarz criterion indicates a betterfitting model for a given number of parameters, or a more parsimonious model for a given goodness of fit.
Null Hypothesis  Maintained Hypothesis  

System  k  SC  VAR(4)  VAR(3)  VAR(2)  
VAR(4)  80 
1111.5 
−29.06 
– – 

↓  –  
VAR(3)  64 
1101.8 
−29.79 
0.82 [0.66] 

↓  (16,128)  
VAR(2)  48 
1091.0 
−30.49 
0.91 [0.61] 
1.02 [0.44] 

↓  (32,156)  (16,141)  
VAR(1)  32 
1079.2 
−31.15 
1.00 [0.48] 
1.11 [0.32] 
1.21 [0.26] 

(48,163)  (32,171)  (16,153)  
Notes 
Having tested for and found weak exogeneity for unit labour costs, import prices, and petrol prices (Table 2), a fourthorder autoregressive distributed lag for the CPI is simplified to the ECM in (17). Table A2 lists the estimates of the coefficients for the fourthorder ADL, where the ADL has been transformed into its unrestricted ECM representation (11). The following variables do not appear either numerically or statistically significant.
Variable 
Lag i  

0  1  2  3  
Δp_{t−i}  −1.0 (–) 
−0.063 (0.141) 
0.136 (0.138) 
−0.043 (0.146) 
Δulc_{t−i}  0.042 (0.031) 
−0.020 (0.028) 
−0.021 (0.029) 
−0.007 (0.028) 
Δip_{t−i}  0.021 (0.016) 
−0.011 (0.019) 
−0.003 (0.021) 
0.000 (0.019) 
Δpet_{t−i}  0.011 (0.007) 
0.006 (0.009) 
0.003 (0.009) 
0.007 (0.008) 
Δy_{t−i}  −0.020 (0.048) 
−0.039 (0.054) 
−0.009 (0.045) 
−0.022 (0.042) 
p_{t−i}  −0.098 (0.034) 

ulc_{t−i}  0.051 (0.028) 

ip_{t−i}  0.046 (0.016) 

pet_{t−i}  0.005 (0.005) 

0.080 (0.027) 

D_{t}  0.011 (0.004) 

S_{it}  −0.0087 (0.0288) 
−0.0021 (0.0013) 
−0.0011 (0.0011) 
−0.0022 (0.0012) 
T = 65 [1977(3)–1993(3)] R^{2} = 0.9016
= 0.268% dw = 2.04 LM_{p} : F(1, 36) = 0.33 AR : F(5, 31) = 0.50 ARCH : F(4, 28) = 0.25 Normality : χ^{2} (2) = 0.34 RESET : F(1, 35) = 0.17 

Notes 
 The third lag on Δp, Δulc, Δip, Δpet, and Δy;
 The second lag on Δp, Δulc, Δip, Δpet, and Δy; and
 The first lag on Δp, Δulc, Δip, Δpet, and Δy.
Three additional sets of reductions are considered:
 The sum of the coefficients on ulc_{t–1}, ip_{t–1}, and pet_{t–1} equals the negative of the coefficient on p_{t–1}, i.e., longrun homogeneity in prices is satisfied;
 Δulc_{t} and Δip_{t} have zero coefficients; and
 Δy_{t} has a zero coefficient.
Treated sequentially, these six restrictions obtain the following seven models.
Model 1:  The unrestricted ECM in Table A2. 
Model 2:  Model 1, excluding the third lag on Δp, Δulc, Δip, Δpet, and Δy. 
Model 3:  Model 2, excluding the second lag on Δp, Δulc, Δip, Δpet, and Δy. 
Model 4:  Model 3, excluding the first lag on Δp, Δulc, Δip, Δpet, and Δy. 
Model 5:  Model 4, imposing longrun price homogeneity. 
Model 6:  Model 5, excluding Δulc_{t} and Δip_{t}. 
Model 7:  Model 6, excluding Δy_{t}. 
So, for example, Model 2 is Model 1 plus reduction (i); Model 3 is Model 1 plus reductions (i)–(ii); and Model 3 is also Model 2 plus reduction (ii).
Table A3 lists the estimates for Model 4 and shows how little the estimates change from imposing the first three reductions. Table A3 also clarifies how the remaining three restrictions appear reasonable.
Variable 
Lag i  

0  1  2  3  
Δp_{t−i}  −1.0 (–) 

Δulc_{t−i}  0.033 (0.022) 

Δip_{t−i}  0.012 (0.013) 

Δpet_{t−i}  0.010 (0.006) 

Δy_{t−i}  −0.049 (0.030) 

p_{t−i}  −0.0884 (0.0136) 

ulc_{t−i}  0.0404 (0.0143) 

ip_{t−i}  0.0417 (0.0055) 

pet_{t−i}  0.0072 (0.0030) 

0.0641 (0.0124) 

D_{t}  0.0103 (0.0028) 

S_{it}  0.0032 (0.0140) 
−0.0022 (0.0010) 
−0.0007 (0.0009) 
−0.0024 (0.0009) 
T = 65 [1977(3)–1993(3)] R^{2}
= 0.8838 = 0.244% dw = 2.13 LM_{p} : F(1, 51) = 0.09 AR : F(5, 46) = 0.27 ARCH : F(4, 43) = 0.95 Normality : χ^{2} (2) = 1.19 RESET : F(1, 50) = 1.00 Hetero : F(22, 28) = 0.92 

Notes 
To facilitate formally assessing whether or not the sequence of reductions (i)–(vi) is valid, and if not, where not, statistics associated with the implied reductions are calculated for all model pairs, and not only for adjacent models. Table A4 reports this information, including the estimated equation standard error and the Schwarz criterion for each model, the F statistics for all model pairs, and the associated tail probability values. The equation standard error is relatively constant across the entire simplification path; and the Schwarz criterion declines steadily through Model 6, remaining virtually unchanged between Model 6 and Model 7. Only that last reduction (excluding Δy_{t}) is statistically significant at the 5% level; and it is only barely so, and only when considered by itself and not in conjunction with previous reductions. Appendix 3 develops an alternative model that includes changes in the output gap. Other orderings of (i)–(vi) generate somewhat different statistics, but those resulting statistics are unlikely to be highly statistically significant because the reduction of (i)–(vi) as a whole appears valid, with F(19, 36) = 0.66 and a pvalue of 0.84.
Null Hypothesis  Maintained Hypothesis (Model Number)  

Model  k  SC  1  2  3  4  5  6  
1  29 
0.268% 
−10.58 
– – 

↓ (i)  –  
2  24 
0.255% 
−10.86 
0.24 [0.94] 

↓ (ii)  (5,36)  
3  19 
0.250% 
−11.11 
0.40 [0.94] 
0.62 [0.69] 

↓ (iii)  (10,36)  (5,41)  
4  14 
0.244% 
−11.37 
0.43 [0.96] 
0.59 [0.82] 
0.58 [0.71] 

↓ (iv)  (15,36)  (10,41)  (5,46)  
5  13 
0.242% 
−11.43 
0.41 [0.97] 
0.54 [0.86] 
0.50 [0.81] 
0.09 [0.77] 

↓ (v)  (16,36)  (11,41)  (6,46)  (1,51)  
6  11 
0.244% 
−11.51 
0.49 [0.95] 
0.65 [0.80] 
0.70 [0.69] 
0.93 [0.43] 
1.37 [0.26] 

↓ (vi)  (18,36)  (13,41)  (8,46)  (3,51)  (2,52)  
7  10 
0.251% 
−11.50 
0.66 [0.84] 
0.89 [0.58] 
1.08 [0.39] 
1.78 [0.15] 
2.38 [0.080] 
4.34 [0.042] 

(19,36)  (14,41)  (9,46)  (4,51)  (3,52)  (1,54)  
Notes 
Figures A1–A3 show the recursively estimated coefficients of the economic variables in (17) and plusorminus twice their recursively estimated standard errors, conventionally denoted and ± 2ese() respectively. To provide more interpretable graphs, (17) has been parameterized as (21), in which all coefficients are unrestricted. Coefficients vary only slightly relative to their ex ante standard errors, and the two dominant feedback terms are highly significant by 1986.