Obtain the inflation surprises and unemployment gap according to Coiban and Gorodnichenko(2015).

Backward-looking inflation expecatation formula:

\[E_t\pi_{t+1}=\frac{1}{4}(\pi_{t-1}+\pi_{t-2}+\pi_{t-3}+\pi_{t-4})\]

Inflation surprises:

\[\pi_t-E_t\pi_{t+1}\]

Unemployment gap:

\[UE_t^{gap}=UE_t-UE_t^{natural}\]

Summary statistics

1985Q1-2007Q3
(N=91)		 2007Q4-2021Q2
(N=55)		 Overall
(N=146)
Inflation surprises
Mean (SD) -0.0576 (1.52) 0.0942 (2.97) -0.000420 (2.18)
Median [Min, Max] -0.0683 [-5.05, 3.48] 0.301 [-14.4, 6.30] -0.0202 [-14.4, 6.30]
Core inflation surprises
Mean (SD) -0.111 (0.565) 0.0862 (1.20) -0.0365 (0.861)
Median [Min, Max] -0.130 [-1.88, 1.22] 0.0154 [-3.36, 6.45] -0.0973 [-3.36, 6.45]
Unemployment gap
Mean (SD) 0.194 (0.840) 1.49 (1.92) 0.683 (1.48)
Median [Min, Max] 0.121 [-1.31, 2.05] 1.24 [-0.937, 8.60] 0.342 [-1.31, 8.60]

Time series plots

Scatter plot to show the Phillips curve relationship

Phillips curve with regular CPI inflation

2008Q4 and 2020Q2 are out of bound so are not reported in the regular CPI inflation graph

Phillips curve with core CPI inflation

OLS and IV estimate of the Phillips curve relationship for the full sample

OLS formula:

\[\pi_t-E_t\pi_{t+1}=c+\kappa UE_t^{gap}+v_t\]

Instrumental variable, two-stage least squares formula:

First stage regression:

\[\widehat{UE_t^{gap}}=\alpha+\beta UE_{t-1}^{gap}+\epsilon_t\]

Outcome regression:

\[\pi_t-E_t\pi_{t+1}=c+\kappa \widehat{UE_{t-1}^{gap}}+u_t\]

Estimates for the models with Newey-West standard errors

OLS Phillips curve model for regular CPI
## 
## t test of coefficients:
## 
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept)  0.045396   0.105265  0.4313   0.6669
## un_gap      -0.067056   0.073274 -0.9151   0.3616
IV Phillips curve model for regular CPI
## 
## t test of coefficients:
## 
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.17357    0.15497 -1.1200   0.2646
## un_gap       0.25953    0.19362  1.3404   0.1823
OLS Phillips curve model for core CPI
## 
## t test of coefficients:
## 
##              Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  0.019087   0.064359  0.2966  0.76722  
## un_gap      -0.081379   0.043547 -1.8688  0.06369 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
IV Phillips curve model for core CPI
## 
## t test of coefficients:
## 
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.088445   0.083607 -1.0579   0.2919
## un_gap       0.081913   0.118317  0.6923   0.4899

Exhaustive search for the split point

Chow-test statistics:

\[C=\frac{RSS-RSS_1-RSS_2}{RSS_1+RSS_2}\times\frac{T-2k}{k}\] \(k\) = number of parameters in the model

Analysis on the regular CPI inflation data

## [1] "Decision: Reject Null of Stability with signif. level 5%"
## [1] "2008-07-01"

Analysis on the core CPI inflation data

## [1] "Decision: Do NOT reject Null of Stability with signif. level 5%"
## [1] "1991-01-01"

Although the Chow test indicates the core CPI inflation series is stable over the entire sample period, I still proceed with the split point to investigate the relationship out of curiosity.

Phillips curve on the regular CPI inflation data with the new split

Phillips curve on the core CPI inflation data with the new split

Seperate IV regression on the 2 sub-periods to find out how much the slope has changed

1985Q1-2008Q2 with rugular CPI inflation
## 
## t test of coefficients:
## 
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept)  0.029977   0.155373  0.1929   0.8474
## un_gap      -0.215351   0.168356 -1.2791   0.2041
2008Q3-2021Q2 with rugular CPI inflation
## 
## t test of coefficients:
## 
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept)  0.055787   0.458390  0.1217   0.9036
## un_gap      -0.042667   0.165770 -0.2574   0.7980
1985Q1-1990Q4 with core CPI inflation
## 
## t test of coefficients:
## 
##              Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  0.065759   0.091314  0.7201  0.47938  
## un_gap      -0.290017   0.137534 -2.1087  0.04715 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
1991Q1-2021Q2 with core CPI inflation
## 
## t test of coefficients:
## 
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.12344    0.10197 -1.2105   0.2285
## un_gap       0.10219    0.12435  0.8218   0.4128

Output the table of the regression results

Table for regular CPI inflation
Dependent variable:
CPI inflation surprises
instrumental OLS instrumental
variable variable
split1 split2 full full
Unemployment gap hat -0.215 -0.043 -0.067 0.260
(0.168) (0.166) (0.073) (0.194)
Constant 0.030 0.056 0.045 -0.174
(0.155) (0.458) (0.105) (0.155)
Observations 93 51 146 145
R2 0.013 0.001 0.002 -0.047
Adjusted R2 0.003 -0.020 -0.005 -0.055
Residual Std. Error 1.531 (df = 91) 3.087 (df = 49) 2.182 (df = 144) 2.243 (df = 143)
F Statistic 0.302 (df = 1; 144)
Note: p<0.1; p<0.05; p<0.01
Table for core CPI inflation
Dependent variable:
Core CPI inflation surprises
instrumental OLS instrumental
variable variable
split1c split2c full full
Unemployment gap hat -0.290** 0.102 -0.081* 0.082
(0.138) (0.124) (0.044) (0.118)
Constant 0.066 -0.123 0.019 -0.088
(0.091) (0.102) (0.064) (0.084)
Observations 23 121 146 145
R2 0.122 -0.078 0.020 -0.059
Adjusted R2 0.080 -0.087 0.013 -0.067
Residual Std. Error 0.606 (df = 21) 0.943 (df = 119) 0.856 (df = 144) 0.891 (df = 143)
F Statistic 2.892* (df = 1; 144)
Note: p<0.1; p<0.05; p<0.01