RDP 9510: Modelling Inflation in Australia Appendix 2: Design of the Empirical ECM

Gordon de Brouwer and Neil R. Ericsson

November 1995

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 mis-specification.

Initially, the vector autoregression for the system cointegration analysis is simplified from a fourth-order VAR to a first-order 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 better-fitting model for a given number of parameters, or a more parsimonious model for a given goodness of fit.

Table A1: F and Related Statistics for the Sequential Reduction from the Fourth-order VAR to the First-order VAR
Null Hypothesis   Maintained Hypothesis
System k Inline Equation 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
1. The first four columns report the vector autoregression with its order, and for that system: the number of unrestricted parameters k, the log-likelihood Inline Equation, and the Schwarz criterion SC.
2. The three entries within a given block of numbers in the last three columns are: the approximate F statistic for testing the null hypothesis (indicated by the model to the left of the entry) against the maintained hypothesis (indicated by the model above the entry), the tail probability associated with that value of the F statistic (in square brackets), and the degrees of freedom for the F statistic (in parentheses). See Doornik and Hendry (1994) for details underlying these calculations.

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

Table A2: The Unrestricted Error Correction Representation for the Underlying CPI
Variable
 		 Lag i
0 1 2 3
Δpt−i −1.0
(–)		 −0.063
(0.141)		 0.136
(0.138)		 −0.043
(0.146)
Δulct−i 0.042
(0.031)		 −0.020
(0.028)		 −0.021
(0.029)		 −0.007
(0.028)
Δipt−i 0.021
(0.016)		 −0.011
(0.019)		 −0.003
(0.021)		 0.000
(0.019)
Δpett−i 0.011
(0.007)		 0.006
(0.009)		 0.003
(0.009)		 0.007
(0.008)
Δyt−i −0.020
(0.048)		 −0.039
(0.054)		 −0.009
(0.045)		 −0.022
(0.042)
pt−i   −0.098
(0.034)		    
ulct−i   0.051
(0.028)		    
ipt−i   0.046
(0.016)		    
pett−i   0.005
(0.005)		    
Inline Equation   0.080
(0.027)		    
Dt 0.011
(0.004)		      
Sit −0.0087
(0.0288)		 −0.0021
(0.0013)		 −0.0011
(0.0011)		 −0.0022
(0.0012)
T = 65 [1977(3)–1993(3)] R2 = 0.9016 Inline Equation = 0.268%
dw = 2.04 LMp : F(1, 36) = 0.33
AR : F(5, 31) = 0.50 ARCH : F(4, 28) = 0.25
Normality : χ2 (2) = 0.34 RESET : F(1, 35) = 0.17

Notes
1. The dependent variable is Δpt. Even so, the equation is in levels, not in differences, noting the presence of pt−1.
2. The variable S0t is the constant term; and S1t, S2t, and S3t are centered seasonal dummies for the first, second, and third quarters, respectively.
  1. The third lag on Δp, Δulc, Δip, Δpet, and Δy;
  2. The second lag on Δp, Δulc, Δip, Δpet, and Δy; and
  3. The first lag on Δp, Δulc, Δip, Δpet, and Δy.

Three additional sets of reductions are considered:

  1. The sum of the coefficients on ulct–1, ipt–1, and pett–1 equals the negative of the coefficient on pt–1, i.e., long-run homogeneity in prices is satisfied;
  2. Δulct and Δipt have zero coefficients; and
  3. Δyt 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 long-run price homogeneity.
Model 6: Model 5, excluding Δulct and Δipt.
Model 7: Model 6, excluding Δyt.

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.

Table A3: A Partially Restricted Error Correction Representation for the Underlying CPI
Variable
 		 Lag i
0 1 2 3
Δpt−i −1.0
(–)		      
Δulct−i 0.033
(0.022)		      
Δipt−i 0.012
(0.013)		      
Δpett−i 0.010
(0.006)		      
Δyt−i −0.049
(0.030)		      
pt−i   −0.0884
(0.0136)		    
ulct−i   0.0404
(0.0143)		    
ipt−i   0.0417
(0.0055)		    
pett−i   0.0072
(0.0030)		    
Inline Equation   0.0641
(0.0124)		    
Dt 0.0103
(0.0028)		      
Sit 0.0032
(0.0140)		 −0.0022
(0.0010)		 −0.0007
(0.0009)		 −0.0024
(0.0009)
T = 65 [1977(3)–1993(3)] R2 = 0.8838 Inline Equation = 0.244%
dw = 2.13 LMp : F(1, 51) = 0.09
AR : F(5, 46) = 0.27 ARCH : F(4, 43) = 0.95
Normality : χ2 (2) = 1.19 RESET : F(1, 50) = 1.00
Hetero : F(22, 28) = 0.92

Notes
See the notes for Table A2.

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 Inline Equation 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 Δyt) 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 p-value of 0.84.

Table A4: F and Related Statistics for the Sequential Reduction from the Fourth-order ADL Model in Table A2
Null Hypothesis   Maintained Hypothesis (Model Number)
Model k Inline Equation 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
1. The first four columns report the model number (with reduction number), and for that model: the number of unrestricted parameters k, the estimated equation standard error Inline Equation, and the Schwarz criterion SC. The text of Appendix 2 defines the models and reductions.
2. The three entries within a given block of numbers in the last six columns are: the F statistic for testing the null hypothesis (indicated by the model number to the left of the entry) against the maintained hypothesis (indicated by the model number above the entry), the tail probability associated with that value of the F statistic (in square brackets), and the degrees of freedom for the F statistic (in parentheses).

Figures A1–A3 show the recursively estimated coefficients of the economic variables in (17) and plus-or-minus twice their recursively estimated standard errors, conventionally denoted Inline Equation and Inline Equation ± 2ese(Inline Equation) 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.

Figure A1: Recursive estimates (—) of the coefficients on Δpett and yres/t−1 with ±2 estimated standard errors (···).
Figure A1: Recursive estimates (—) of the coefficients on Δpett and with ±2 estimated standard errors (···).
Figure A2: Recursive estimates (—) of the coefficients on (p – ulc)t−1 and (p – ip)t−1, with ±2 estimated standard errors (···).
Figure A2: Recursive estimates (—) of the coefficients on (p – ulc)t–1 and (p – ip)t–1, with ±2 estimated standard errors (···).
Figure A3: Recursive estimates (—) of the coefficients on (p – pet)t−1 and the constant, with ±2 estimated standard errors (···).
Figure A3: Recursive estimates (—) of the coefficients on (p – pet)t–1 and the constant, with ±2 estimated standard errors (···).