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Core Inflation Rate for China and the ASEAN-10 Countries: Smoothed Signal for Score-Driven Local Level Plus Scale Models

  • Szabolcs Blazsek , Adrian Licht , Astrid Ayala and Su-Ping Liu ORCID logo EMAIL logo
Published/Copyright: January 1, 2024
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Studies in Nonlinear Dynamics & Econometrics
From the journal Studies in Nonlinear Dynamics & Econometrics Volume 29 Issue 2

Abstract

We use a score-driven minimum mean-squared error (MSE) signal extraction method and perform inflation smoothing for China and the ASEAN-10 countries. Our focus on China and ASEAN-10 countries is motivated by the significant historical variation in inflation rates, e.g. during the 1997 Asian Financial Crisis, the 2007–2008 Financial Crisis, the COVID-19 Pandemic, and the Russian Invasion of Ukraine. Some advantages of the score-driven signal extraction method are that it uses dynamic mean and volatility filters, it considers stationary or non-stationary mean dynamics, it is computationally fast, it is robust to extreme observations, it uses information-theoretically optimal updating mechanisms for both mean and volatility, it uses closed-form formulas for smoothed signals, and parameters are estimated by using the maximum likelihood (ML) method for which the asymptotic properties of estimates are known. In the empirical application, we present the political and economic conditions for each country and analyze the evolution and determinants of the core inflation rate.

Keywords: core inflation rate; minimum mean-squared error (MSE) signal extraction; generalized autoregressive score (GAS); dynamic conditional score (DCS); monetary policy
JEL Classification: C22; C52; E31; E52; E58
Corresponding author: Su-Ping Liu, College of Business and Public Management, Wenzhou-Kean University, 88 Daxue Rd, Ouhai, Wenzhou, 325060, Zhejiang Province, China, E-mail:

Funding source: Wenzhou-Kean University

Acknowledgment

We acknowledge financial assistance from the Wenzhou-Kean University (International Collaborative Research Programs, ICRP202207). We are thankful for the comments of Juan Carlos Castañeda Fuentes and Luis Albériko Gil Alana. All remaining errors are our own. Data and computer codes are available from the authors upon request. Declarations of interest: none.

Appendix A

We assume that f ( y t | F t − 1 , Θ ) is a correctly specified conditional density (Wooldridge 1994). Asymptotically at the true values of parameters, the scaled score function with respect to location l t , and the score function with respect to log-scale z t have the following properties: (i)

(A.1) E t − 1 ∂ ⁡ ln ⁡ f ( y t | F t − 1 , Θ ) ∂ Θ ′ = E t − 1 ∂ ⁡ ln ⁡ f ( y t | F t − 1 , Θ ) ∂ s t × ∂ s t ∂ Θ ′ = 0 ,

where index t − 1 indicates expectations that are conditional on F t − 1 . Since ∂s t /∂Θ′ ≠ 0,

(A.2) E t − 1 ∂ ⁡ ln ⁡ f ( y t | F t − 1 , Θ ) ∂ s t = E t − 1 ν + 1 ν ⁡ exp ( 2 λ t ) l t = E t − 1 ( l t ) ν + 1 ν ⁡ exp ( 2 λ t ) = 0 .

As a consequence, Et−1(l t ) = 0 (i.e. l t is an MDS). Moreover,

(A.3) E t − 1 ∂ ⁡ ln ⁡ f ( y t | F t − 1 , Θ ) ∂ Θ ′ = E t − 1 ∂ ⁡ ln ⁡ f ( y t | F t − 1 , Θ ) ∂ λ t × ∂ λ t ∂ Θ ′ = 0 .

Since ∂λ t /∂Θ′ ≠ 0,

(A.4) E t − 1 ∂ ⁡ ln ⁡ f ( y t | F t − 1 , Θ ) ∂ λ t = E t − 1 ( z t ) = 0 .

Thus, z t is an MDS. (ii) E(l t ) = 0 and E(z t ) = 0, due to the law of iterated expectations. (iii) l t and z t are contemporaneously correlated, as both are functions of v t . (iv) Scaled score function l t is not i.i.d., as it depends on λ t . (v) We assume that |λ t | < λmax < ∞ for all t (Blazsek, Escribano, and Licht 2024), which sets an exogenous bound for dynamic scale. The consequence of this assumption is that l t is a bounded function of ϵ t (Blazsek, Escribano, and Licht 2024). Therefore, Var(l t ) < ∞, and l t is white noise. If the roots of 1 − ϕ 1 z − ⋯ − ϕ p z p = 0 lie outside the unit circle, then s t is covariance stationary for (10). (vi) z t is a bounded function of ϵ t (Harvey 2013). Therefore, Var(z t ) < ∞, and z t is white noise. If |b| < 1, then λ t is covariance stationary. (vii) Due to |λ t | < ∞, ∂l t /∂λ t and ∂z t /∂s t are bounded functions of ϵ t (Blazsek, Escribano, and Licht 2024). (viii) ∂l t /∂s t and ∂z t /∂λ t are bounded functions of ϵ t (Blazsek, Escribano, and Licht 2024). (ix) Scaled score function l t is an F -measurable function of ϵ t (White 2001), because l t is a continuous function of ϵ t . Scaled score function l t is strictly stationary and ergodic, because l t is an F -measurable function of (ϵ1, …, ϵ t ), and because ϵ t is strictly stationary and ergodic (White 2001). (x) Score function z t is i.i.d., because z t is a continuous function of ϵ t , and because ϵ t is i.i.d. (White 2001). (xi) Score function z t is an F -measurable function of ϵ t (White 2001), because z t is a continuous function of ϵ t (Harvey 2013). Score function z t is strictly stationary and ergodic, because z t is an F -measurable function of ϵ t , and because ϵ t is strictly stationary and ergodic (White 2001).

Appendix B

The following results are valid asymptotically at the true values of parameters, under the assumption of correct model specification. Matrix CU,V with dimensions (T − p) × T is represented as:

(B.1) C U , V = B D 2 … … … … D T C 0 B ⋱ ⋱ ⋱ ⋱ ⋮ C 1 C 0 B ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋱ ⋱ ⋮ C J … C 1 C 0 B D 2 ⋮ = 0 0 … … … … 0 C 0 0 ⋱ ⋱ ⋱ ⋱ 0 0 C 0 0 ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋱ ⋱ ⋮ 0 … 0 C 0 0 … 0 ,

where J = (T − p − 2). In the following, we formulate the elements B, C0, C i for i = 1, …, J, and D j for j = 2, …, T. First, B = Cov(ψ1lt−1, v t ) = Cov[ψ1lt−1, exp(λ t )ϵ t ] = 0, because v t is an MDS:

(B.2) E ( v t | F t − 1 ) = E [ exp ( λ t ) ϵ t | F t − 1 ] = exp ( λ t ) E [ ϵ t | F t − 1 ] = 0 .

Therefore, E ( l t − 1 v t | F t − 1 ) = l t − 1 E ( v t | F t − 1 ) = 0 , hence, E(v t lt−1) = 0. Second, C0 = Cov(ψ1l t , v t ) = ψ1νE[exp(2λ t )]/(ν + 1), where Eqs. (19) and (20) are used for the computation of E[exp(2λ t )]. Third, C i = Cov[ψ1lt+i, exp(λ t )ϵ t ] = E[ψ1lt+i exp(λ t )ϵ t ] for 1 ≤ i ≤ J, for which we use:

(B.3) E [ ψ 1 l t + i ⁡ exp ( λ t ) ϵ t | F t + i − 1 ] = ψ 1 E [ l t + i | F t + i − 1 ] ︸ 0 exp ( λ t ) ϵ t = 0 ,

which is true because l t is an MDS. Hence, C i = 0 due to the law of iterated expectations. Fourth, D j = Cov[ψ1lt−j, exp(λ t )ϵ t ] = 0 for j = 2, …, T, because of the following arguments. We use:

(B.4) E [ ψ 1 l t − j ⁡ exp ( λ t ) ϵ t | F t − j ] = ψ 1 l t − j E [ exp ( λ t ) ϵ t | F t − j ]

and we use the law of iterated expectations (White 2001) as follows:

(B.5) E [ exp ( λ t ) ϵ t | F t − j ] = E { E [ exp ( λ t ) ϵ t | F t − 1 ] | F t − j } = E { exp ( λ t ) E [ ϵ t | F t − 1 ] ︸ 0 | F t − j } = 0 ,

which holds because E(|v t |) < ∞ (White 2001). We use Eq. (B.5), and the law of iterated expectations for Eq. (B.4), and we obtain that D j = 0.

Appendix C
Table C1:

Descriptive statistics of the inflation rate and maximum likelihood estimates for score-driven models for Indonesia.

(a). Descriptive statistics of inflation rate (% points)
Start date January 1969 Standard deviation 8.6165
End date February 2023 Skewness 2.8720
Sample size, T 650 Excess kurtosis 10.6919
Mean 9.6459 Shapiro–Wilk test statistic (p-value) 0.7115***(0.0000)
Median 7.5716 ADF test statistic, constant (p-value) −3.6622***(0.0047)
Minimum −5.2843 Local whittle estimator (standard error) 0.7597(0.0729)
Maximum 60.1117 ARCH test statistic (p-value) 621.8560***(0.0000)
(b). Local level s t with |ϕ1| < 1; scale λ t = λ (c). Local level s t with ϕ1 = 1; scale λ t = λ
ν 2.8882***(0.1189) ν 1.9061***(0.0719)
λ 0.4439***(0.0322) λ 0.2815***(0.0284)
ϕ 1 0.9168***(0.0041) ϕ 1
ψ 1 1.9224***(0.1028) ψ 1 2.0751***(0.1026)
LL −2.2317 LL −2.2696
AIC 4.4757 AIC 4.5485
BIC 4.5033 BIC 4.5692
HQC 4.4864 HQC 4.5565
(d). Local level s t with |ϕ1| < 1; scale λ t (e). Local level s t with ϕ1 = 1; scale λ t
ν 17.1222***(1.5817) ν 17.5558***(1.6152)
a 0.0144***(0.0038) a 0.0142***(0.0038)
b 0.9927***(0.0020) b 0.9928***(0.0020)
c 0.1104***(0.0076) c 0.1099***(0.0075)
ϕ 1 0.9921***(0.0052) ϕ 1
ψ 1 1.4785***(0.0585) ψ 1 1.4822***(0.0588)
LL −1.5178 LL −1.5194
AIC 3.0541 AIC 3.0542
BIC 3.0954 BIC 3.0887
HQC 3.0701 HQC 3.0676

  1. Augmented Dickey–Fuller (ADF); autoregressive conditional heteroskedasticity (ARCH); ordinary least squares (OLS); heteroskedasticity and autocorrelation consistent (HAC); log-likelihood (LL); Akaike information criterion (AIC); Bayesian information criterion (BIC); Hannan–Quinn criterion (HQC). For ADF, lag-order is selected using BIC. For ARCH, lag-order is selected using the partial autocorrelation function (PACF) of squared-inflation. Standard errors of parameters are in parentheses. *** is significance at the 1 % levels. Best specifications are shown by bold numbers.

Table C2:

Descriptive statistics of the inflation rate and maximum likelihood estimates for score-driven models for the Philippines.

(a). Descriptive statistics of inflation rate (% points)
Start date January 2011 Standard deviation 1.5905
End date November 2022 Skewness 0.3176
Sample size, T 143 Excess kurtosis 0.2349
Mean 3.1178 Shapiro–Wilk test statistic (p-value) 0.9856+(0.1425)
Median 2.9961 ADF test statistic, constant (p-value) −2.1683(0.2182)
Minimum −0.3735 Local whittle estimator (standard error) 0.9428(0.1179)
Maximum 7.6827 ARCH test statistic (p-value) 128.9340***(0.0000)
(b). Local level s t with |ϕ1| < 1; scale λ t = λ (c). Local level s t with ϕ1 = 1; scale λ t = λ
ν 6.1780***(1.7443) ν 6.1445***(1.7198)
λ −0.8069***(0.0706) λ −0.8056***(0.0703)
ϕ 1 0.9733***(0.0408) ϕ 1
ψ 1 1.8733***(0.2038) ψ 1 1.8967***(0.1996)
LL −0.7797 LL −0.7820
AIC 1.6153 AIC 1.6059
BIC 1.6982 BIC 1.6680
HQC 1.6490 HQC 1.6311
(d). Local level s t with |ϕ1| < 1; scale λ t (e). Local level s t with ϕ1 = 1; scale λ t
ν 4.6788***(1.1803) ν 8.9930***(1.4040)
a 0.0011(0.0027) a 0.0007(0.0021)
b 0.9898***(0.0143) b 0.9925***(0.0125)
c 0.0982**(0.0392) c 0.0837**(0.0352)
ϕ 1 0.9560***(0.0393) ϕ 1
ψ 1 1.5496***(0.1801) ψ 1 1.5648***(0.1858)
LL −0.5642 LL −0.5714
AIC 1.2124 AIC 1.2127
BIC 1.3367 BIC 1.3162
HQC 1.2629 HQC 1.2547

  1. Augmented Dickey–Fuller (ADF); autoregressive conditional heteroskedasticity (ARCH); ordinary least squares (OLS); heteroskedasticity and autocorrelation consistent (HAC); log-likelihood (LL); Akaike information criterion (AIC); Bayesian information criterion (BIC); Hannan–Quinn criterion (HQC). For ADF, lag-order is selected using BIC. For ARCH, lag-order is selected using the partial autocorrelation function (PACF) of squared-inflation. Standard errors of parameters are in parentheses. +, **, and *** are significance at the 15 %, 5 %, and 1 % levels, respectively. Best specifications are shown by bold numbers.

Table C3:

Descriptive statistics of the inflation rate and maximum likelihood estimates for score-driven models for Thailand.

(a). Descriptive statistics of inflation rate (% points)
Start date January 2011 Standard deviation 2.0302
End date January 2023 Skewness 0.6555
Sample size, T 145 Excess kurtosis 0.6381
Mean 1.5830 Shapiro–Wilk test statistic (p-value) 0.9666***(0.0013)
Median 1.2485 ADF test statistic, constant (p-value) −1.8542(0.3545)
Minimum −3.4967 Local whittle estimator (standard error) 0.9088(0.1179)
Maximum 7.5655 ARCH test statistic (p-value) 109.8050***(0.0000)
(b). Local level s t with |ϕ1| < 1; scale λ t = λ (c). Local level s t with ϕ1 = 1; scale λ t = λ
ν 3.2885***(0.6289) ν 4.6179***(1.0368)
λ −0.4656***(0.0645) λ −0.3608***(0.0575)
ϕ 1 0.9503***(0.0228) ϕ 1
ψ 1 1.7278***(0.2846) ψ 1 1.6235***(0.2596)
LL −1.2755 LL −1.2846
AIC 2.6062 AIC 2.6106
BIC 2.6883 BIC 2.6722
HQC 2.6396 HQC 2.6357
(d). Local level s t with |ϕ1| < 1; scale λ t (e). Local level s t with ϕ1 = 1; scale λ t
ν no convergence ν no convergence
a a
b b
c c
ϕ 1 ϕ 1
ψ 1 ψ 1
LL LL
AIC AIC
BIC BIC
HQC HQC

  1. Augmented Dickey–Fuller (ADF); autoregressive conditional heteroskedasticity (ARCH); ordinary least squares (OLS); heteroskedasticity and autocorrelation consistent (HAC); log-likelihood (LL); Akaike information criterion (AIC); Bayesian information criterion (BIC); Hannan–Quinn criterion (HQC). For ADF, lag-order is selected using BIC. For ARCH, lag-order is selected using the partial autocorrelation function (PACF) of squared-inflation. Standard errors of parameters are in parentheses. *** is significance at the 1 % level. Best specifications are shown by bold numbers. For the score-driven local level and scale model, the numerical optimization procedure of the LL did not converge, due to model misspecification.

Table C4:

Descriptive statistics of the inflation rate and maximum likelihood estimates for score-driven models for Malaysia.

(a). Descriptive statistics of inflation rate (% points)
Start date January 2001 Standard deviation 1.6406
End date December 2022 Skewness 0.1618
Sample size, T 264 Excess kurtosis 2.6097
Mean 2.0786 Shapiro–Wilk test statistic (p-value) 0.9428***(0.0000)
Median 1.9813 ADF test statistic, constant (p-value) −4.8005***(0.0000)
Minimum −2.9328 Local whittle estimator (standard error) 0.7836(0.0962)
Maximum 8.1715 ARCH test statistic (p-value) 210.0840***(0.0000)
(b). Local level s t with |ϕ1| < 1; scale λ t = λ (c). Local level s t with ϕ1 = 1; scale λ t = λ
ν 6.5056***(21.3843) ν 3.3384***(0.3028)
λ −0.5444***(0.0301) λ −0.6813***(0.0435)
ϕ 1 0.8851***(0.0368) ϕ 1
ψ 1 1.4121***(0.0784) ψ 1 1.8457***(0.1506)
LL −0.8861 LL −1.0548
AIC 1.8025 AIC 2.1323
BIC 1.8567 BIC 2.1729
HQC 1.8243 HQC 2.1486
(d). Local level s t with |ϕ1| < 1; scale λ t (e). Local level s t with ϕ1 = 1; scale λ t
ν 5.3247***(0.6405) ν 6.1672***(0.8281)
a −0.1689***(0.0633) a −0.1533**(0.0649)
b 0.7358***(0.0608) b 0.7455***(0.0686)
c 0.3650***(0.0374) c 0.3358***(0.0389)
ϕ 1 0.8918***(0.0322) ϕ 1
ψ 1 1.6551***(0.0990) ψ 1 1.7062***(0.0850)
LL −0.7449 LL −0.7777
AIC 1.5352 AIC 1.5932
BIC 1.6165 BIC 1.6609
HQC 1.5679 HQC 1.6204

  1. Augmented Dickey–Fuller (ADF); autoregressive conditional heteroskedasticity (ARCH); ordinary least squares (OLS); heteroskedasticity and autocorrelation consistent (HAC); log-likelihood (LL); Akaike information criterion (AIC); Bayesian information criterion (BIC); Hannan–Quinn criterion (HQC). For ADF, lag-order is selected using BIC. For ARCH, lag-order is selected using the partial autocorrelation function (PACF) of squared-inflation. Standard errors of parameters are in parentheses. ** and *** are significance at the 5 % and 1 % levels, respectively. Best specifications are shown by bold numbers.

Table C5:

Descriptive statistics of the inflation rate and maximum likelihood estimates for score-driven models for Vietnam.

(a). Descriptive statistics of inflation rate (% points)
Start date December 2002 Standard deviation 5.2413
End date December 2022 Skewness 1.6662
Sample size, T 241 Excess kurtosis 2.6478
Mean 6.3474 Shapiro–Wilk test statistic (p-value) 0.8283***(0.0000)
Median 4.7538 ADF test statistic, constant (p-value) −2.4169(0.1370)
Minimum −0.9760 Local whittle estimator (standard error) 1.2811(0.1000)
Maximum 24.9340 ARCH test statistic (p-value) 233.0320***(0.0000)
(b). Local level s t with |ϕ1| < 1; scale λ t = λ (c). Local level s t with ϕ1 = 1; scale λ t = λ
ν 14.1527***(3.6718) ν 14.5319***(3.7083)
λ −0.0620(0.0427) λ −0.0565(0.0419)
ϕ 1 0.9793***(0.0131) ϕ 1
ψ 1 1.8171***(0.1280) ψ 1 1.8423***(0.1264)
LL −1.4288 LL −1.4323
AIC 2.8908 AIC 2.8896
BIC 2.9487 BIC 2.9329
HQC 2.9141 HQC 2.9070
(d). Local level s t with |ϕ1| < 1; scale λ t (e). Local level s t with ϕ1 = 1; scale λ t
ν 15.9126***(5.1628) ν 13.8547***(3.6895)
a 0.0244**(0.0113) a 0.0241**(0.0113)
b 0.9734***(0.0120) b 0.9735***(0.0123)
c 0.1847***(0.0306) c 0.1886***(0.0327)
ϕ 1 0.9702***(0.0165) ϕ 1
ψ 1 1.6853***(0.1144) ψ 1 1.7508***(0.1112)
LL −1.0879 LL −1.0945
AIC 2.2256 AIC 2.2304
BIC 2.3123 BIC 2.3027
HQC 2.2605 HQC 2.2595

  1. Augmented Dickey–Fuller (ADF); autoregressive conditional heteroskedasticity (ARCH); ordinary least squares (OLS); heteroskedasticity and autocorrelation consistent (HAC); log-likelihood (LL); Akaike information criterion (AIC); Bayesian information criterion (BIC); Hannan–Quinn criterion (HQC). For ADF, lag-order is selected using BIC. For ARCH, lag-order is selected using the partial autocorrelation function (PACF) of squared-inflation. Standard errors of parameters are in parentheses. ** and *** are significance at the 5 % and 1 % levels, respectively. Best specifications are shown by bold numbers.

Table C6:

Descriptive statistics of the inflation rate and maximum likelihood estimates for score-driven models for China.

(a). Descriptive statistics of inflation rate (% points)
Start date January 1994 Standard deviation 4.8578
End date February 2023 Skewness 2.6006
Sample size, T 350 Excess kurtosis 7.1513
Mean 3.2673 Shapiro–Wilk test statistic (p-value) 0.6905***(0.0000)
Median 1.9322 ADF test statistic, constant (p-value) −4.5653***(0.0001)
Minimum −2.2246 Local whittle estimator (standard error) 1.2603(0.0884)
Maximum 24.4514 ARCH test statistic (p-value) 344.6480***(0.0000)
(b). Local level s t with |ϕ1| < 1; scale λ t = λ (c). Local level s t with ϕ1 = 1; scale λ t = λ
ν 1.8036***(0.0926) ν 1.9179***(0.0948)
λ −0.3932***(0.0438) λ −0.3556***(0.0398)
ϕ 1 ▶1.0000***(0.0111) ϕ 1
ψ 1 2.1170***(0.1472) ψ 1 2.1210***(0.1398)
LL −1.6287 LL −1.6288
AIC 3.2802 AIC 3.2748
BIC 3.3243 BIC 3.3079
HQC 3.2978 HQC 3.2880
(d). Local level s t with |ϕ1| < 1; scale λ t (e). Local level s t with ϕ1 = 1; scale λ t
ν 6.4868***(1.5919) ν 6.3035***(1.4276)
a 0.0191(0.0136) a 0.0162(0.0125)
b 0.9928***(0.0050) b 0.9939***(0.0046)
c 0.2310***(0.0295) c 0.2178***(0.0293)
ϕ 1 0.9689***(0.0109) ϕ 1
ψ 1 1.3030***(0.1099) ψ 1 1.3543***(0.1086)
LL −1.1267 LL −1.1349
AIC 2.2878 AIC 2.2985
BIC 2.3539 BIC 2.3536
HQC 2.3141 HQC 2.3204

  1. Augmented Dickey–Fuller (ADF); autoregressive conditional heteroskedasticity (ARCH); ordinary least squares (OLS); heteroskedasticity and autocorrelation consistent (HAC); log-likelihood (LL); Akaike information criterion (AIC); Bayesian information criterion (BIC); Hannan–Quinn criterion (HQC). For ADF, lag-order is selected using BIC. For ARCH, lag-order is selected using the partial autocorrelation function (PACF) of squared-inflation. Standard errors of parameters are in parentheses. *** is significance at the 1 % level. Best specifications are shown by bold numbers. Misspecified models are shown by ▶.

Table C7:

Descriptive statistics of the inflation rate and maximum likelihood estimates for score-driven models for Cambodia.

(a). Descriptive statistics of inflation rate (% points)
Start date January 2002 Standard deviation 4.9606
End date January 2023 Skewness 2.7977
Sample size, T 253 Excess kurtosis 10.9554
Mean 4.1542 Shapiro–Wilk test statistic (p-value) 0.7106***(0.0000)
Median 3.0451 ADF test statistic, constant (p-value) −3.2820**(0.0157)
Minimum −6.8898 Local whittle estimator (standard error) 0.9961(0.0981)
Maximum 30.4333 ARCH test statistic (p-value) 238.7990***(0.0000)
(b). Local level s t with |ϕ1| < 1; scale λ t = λ (c). Local level s t with ϕ1 = 1; scale λ t = λ
ν 1.8424***(0.1534) ν 1.4376***(0.1013)
λ 0.0087(0.0539) λ −0.1281***(0.0455)
ϕ 1 0.9067***(0.0112) ϕ 1
ψ 1 1.9982***(0.2021) ψ 1 2.2889***(0.1767)
LL −2.0174 LL −2.0545
AIC 4.0664 AIC 4.1328
BIC 4.1223 BIC 4.1747
HQC 4.0889 HQC 4.1496
(d). Local level s t with |ϕ1| < 1; scale λ t (e). Local level s t with ϕ1 = 1; scale λ t
ν 16.0896***(5.5067) ν 15.5721***(5.0544)
a 0.0090(0.0068) a 0.0089(0.0067)
b 0.9916***(0.0062) b 0.9917***(0.0062)
c 0.0934***(0.0178) c 0.0921***(0.0180)
ϕ 1 0.9300***(0.0332) ϕ 1
ψ 1 1.3405***(0.1007) ψ 1 1.4024***(0.1007)
LL −1.4271 LL −1.4400
AIC 2.9016 AIC 2.9196
BIC 2.9854 BIC 2.9894
HQC 2.9353 HQC 2.9477

  1. Augmented Dickey–Fuller (ADF); autoregressive conditional heteroskedasticity (ARCH); ordinary least squares (OLS); heteroskedasticity and autocorrelation consistent (HAC); log-likelihood (LL); Akaike information criterion (AIC); Bayesian information criterion (BIC); Hannan–Quinn criterion (HQC). For ADF, lag-order is selected using BIC. For ARCH, lag-order is selected using the partial autocorrelation function (PACF) of squared-inflation. Standard errors of parameters are in parentheses. ** and *** are significance at the 5 % and 1 % levels, respectively. Best specifications are shown by bold numbers.

Table C8:

Descriptive statistics of the inflation rate and maximum likelihood estimates for score-driven models for Laos.

(a). Descriptive statistics of inflation rate (% points)
Start date January 2006 Standard deviation 5.513800447
End date January 2023 Skewness 3.392788896
Sample size, T 205 Excess kurtosis 13.87769903
Mean 3.1357 Shapiro–Wilk test statistic (p-value) 0.6432***(0.0000)
Median 4.2634 ADF test statistic, constant (p-value) −0.4357(0.9008)
Minimum −1.8425 Local whittle estimator (standard error) 0.8211(0.1043)
Maximum 33.8629 ARCH test statistic (p-value) 199.0360***(0.0000)
(b). Local level s t with |ϕ1| < 1; scale λ t = λ (c). Local level s t with ϕ 1 = 1; scale λ t = λ
ν 2.1870***(0.1321) ν 2.6026***(0.1839)
λ −0.3376***(0.0333) λ −0.2390***(0.0574)
ϕ 1 0.9735***(0.0021) ϕ 1
ψ 1 2.5369***(0.0513) ψ 1 2.2698***(0.1439)
LL −1.5742 LL −1.5911
AIC 3.1875 AIC 3.2115
BIC 3.2523 BIC 3.2601
HQC 3.2137 HQC 3.2312
(d). Local level s t with |ϕ1| < 1; scale λ t . (e). Local level s t with ϕ1 = 1; scale λ t
ν 15.4221***(5.1120) ν 15.6714***(5.1087)
a 0.0078(0.0052) a 0.0078(0.0053)
b 0.9811***(0.0121) b 0.9810***(0.0123)
c 0.1125***(0.0145) c 0.1113***(0.0143)
ϕ 1 0.9838***(0.0155) ϕ 1
ψ 1 1.5712***(0.1333) ψ 1 1.5796***(0.1240)
LL −1.0823 LL −1.0854
AIC 2.2232 AIC 2.2196
BIC 2.3205 BIC 2.3006
HQC 2.2625 HQC 2.2523

  1. Augmented Dickey–Fuller (ADF); autoregressive conditional heteroskedasticity (ARCH); ordinary least squares (OLS); heteroskedasticity and autocorrelation consistent (HAC); log-likelihood (LL); Akaike information criterion (AIC); Bayesian information criterion (BIC); Hannan–Quinn criterion (HQC). For ADF, lag-order is selected using BIC. For ARCH, lag-order is selected using the partial autocorrelation function (PACF) of squared-inflation. Standard errors of parameters are in parentheses. *** is significance at the 1 % levels. Best specifications are shown by bold numbers.

Table C9:

Descriptive statistics of the inflation rate and maximum likelihood estimates for score-driven models for Myanmar.

(a). Descriptive statistics of inflation rate (% points)
Start date January 2011 Standard deviation 2.9778
End date November 2020 Skewness −0.1211
Sample size, T 119 Excess kurtosis 0.3889
Mean 5.6143 Shapiro–Wilk test statistic (p-value) 0.9857(0.2416)
Median 5.7073 ADF test statistic, constant (p-value) −3.1611**(0.0224)
Minimum −2.0644 Local whittle estimator (standard error) 0.6138(0.1250)
Maximum 13.4203 ARCH test statistic (p-value) 97.5355***(0.0000)
(b). Local level s t with |ϕ1| < 1; scale λ t = λ (c). Local level s t with ϕ1 = 1; scale λ t = λ
ν 12.9378***(3.4705) ν 12.4009***(3.2739)
λ 0.2459***(0.0550) λ 0.2575***(0.0562)
ϕ 1 0.8710***(0.0949) ϕ 1
ψ 1 1.6429***(0.2110) ψ 1 1.7422***(0.1932)
LL −1.7435 LL −1.7586
AIC 3.5543 AIC 3.5676
BIC 3.6477 BIC 3.6377
HQC 3.5922 HQC 3.5961
(d). Local level s t with |ϕ1| < 1; scale λ t (e). Local level s t with ϕ1 = 1; scale λ t
ν 10.8384**(4.3829) ν 13.1997*(6.2291)
a 0.0281*(0.0167) a 0.0237(0.0168)
b 0.9704***(0.0179) b 0.9760***(0.0171)
c 0.1256***(0.0343) c 0.1201***(0.0328)
ϕ 1 0.8593***(0.0824) ϕ 1
ψ 1 1.5802***(0.1649) ψ 1 1.6405***(0.1424)
LL −1.5476 LL −1.5719
AIC 3.1960 AIC 3.2278
BIC 3.3362 BIC 3.3446
HQC 3.2529 HQC 3.2752

  1. Augmented Dickey–Fuller (ADF); autoregressive conditional heteroskedasticity (ARCH); ordinary least squares (OLS); heteroskedasticity and autocorrelation consistent (HAC); log-likelihood (LL); Akaike information criterion (AIC); Bayesian information criterion (BIC); Hannan–Quinn criterion (HQC). For ADF, lag-order is selected using BIC. For ARCH, lag-order is selected using the partial autocorrelation function (PACF) of squared-inflation. Standard errors of parameters are in parentheses. *, **, and *** are significance at the 10 %, 5 %, and 1 % levels, respectively. Best specifications are shown by bold numbers.

Table C10:

Descriptive statistics of the inflation rate and maximum likelihood estimates for score-driven models for Brunei.

(a). Descriptive statistics of inflation rate (% points)
Start date January 2011 Standard deviation 1.3403
End date December 2022 Skewness 0.9352
Sample size, T 144 Excess kurtosis 0.4846
Mean 0.5230 Shapiro–Wilk test statistic (p-value) 0.9221***(0.0000)
Median 0.1274 ADF test statistic, constant (p-value) −1.7083(0.4271)
Minimum −1.6900 Local whittle estimator (standard error) 1.0154(0.1179)
Maximum 4.3820 ARCH test statistic (p-value) 121.0650***(0.0000)
(b). Local level s t with |ϕ1| < 1; scale λ t = λ (c). Local level s t with ϕ1 = 1; scale λ t = λ
ν 17.2280***(5.9846) ν 19.9494***(6.7006)
λ −0.8102***(0.0609) λ −0.7946***(0.0525)
ϕ 1 0.9603***(0.0287) ϕ 1
ψ 1 0.9931***(0.1361) ψ 1 0.9845***(0.1308)
LL −0.6676 LL −0.6750
AIC 1.3907 AIC 1.3917
BIC 1.4732 BIC 1.4536
HQC 1.4243 HQC 1.4169
(d). Local level s t with |ϕ1| < 1; scale λ t (e). Local level s t with ϕ1 = 1; scale λ t
ν 10.0124*(5.1364) ν 10.1040**(4.9730)
a −0.5409***(0.1979) a −0.5555***(0.2015)
b 0.4076**(0.1913) b 0.3897**(0.1939)
c 0.2351***(0.0571) c 0.2330***(0.0566)
ϕ 1 0.9892***(0.0189) ϕ 1
ψ 1 0.8737***(0.1600) ψ 1 0.8748***(0.1572)
LL −0.6126 LL −0.6136
AIC 1.3086 AIC 1.2966
BIC 1.4323 BIC 1.3997
HQC 1.3589 HQC 1.3385

  1. Augmented Dickey–Fuller (ADF); autoregressive conditional heteroskedasticity (ARCH); ordinary least squares (OLS); heteroskedasticity and autocorrelation consistent (HAC); log-likelihood (LL); Akaike information criterion (AIC); Bayesian information criterion (BIC); Hannan–Quinn criterion (HQC). For ADF, lag-order is selected using BIC. For ARCH, lag-order is selected using the partial autocorrelation function (PACF) of squared-inflation. Standard errors of parameters are in parentheses. *, **, and *** are significance at the 10 %, 5 %, and 1 % levels, respectively. Best specifications are shown by bold numbers.

Table C11:

Descriptive statistics of the inflation rate and maximum likelihood estimates for score-driven models for Singapore.

(a). Descriptive statistics of inflation rate (% points)
Start date January 2006 Standard deviation 2.3603
End date February 2023 Skewness 0.6525
Sample size, T 206 Excess kurtosis −0.8501
Mean 2.0599 Shapiro–Wilk test statistic (p-value) 0.9036***(0.0000)
Median 1.1640 ADF test statistic, constant (p-value) −1.9239(0.3215)
Minimum −1.5787 Local whittle estimator (standard error) 1.3664(0.1042)
Maximum 7.2990 ARCH test statistic (p-value) 171.2860***(0.0000)
(b). Local level s t with |ϕ1| < 1; scale λ t = λ (c). Local level s t with ϕ1 = 1; scale λ t = λ
ν 11.2133***(2.8797) ν 10.9047***(2.8725)
λ −0.5437***(0.0512) λ −0.5423***(0.0524)
ϕ 1 0.9734***(0.0233) ϕ 1
ψ 1 1.3677***(0.1440) ψ 1 1.3857***(0.1459)
LL −0.9663 LL −0.9703
AIC 1.9715 AIC 1.9697
BIC 2.0361 BIC 2.0182
HQC 1.9976 HQC 1.9893
(d). Local level s t with |ϕ1| < 1; scale λ t (e). Local level s t with ϕ1 = 1; scale λ t
ν 9.4605**(3.7195) ν 9.4092**(3.7362)
a 0.0010(0.0019) a 0.0010(0.0018)
b 0.9945***(0.0083) b 0.9946***(0.0079)
c 0.0845***(0.0255) c 0.0858***(0.0235)
ϕ 1 0.9932***(0.0148) ϕ 1
ψ 1 1.3446***(0.1744) ψ 1 1.3511***(0.1749)
LL −0.8115 LL −0.8120
AIC 1.6813 AIC 1.6726
BIC 1.7782 BIC 1.7533
HQC 1.7205 HQC 1.7052

  1. Augmented Dickey–Fuller (ADF); autoregressive conditional heteroskedasticity (ARCH); ordinary least squares (OLS); heteroskedasticity and autocorrelation consistent (HAC); log-likelihood (LL); Akaike information criterion (AIC); Bayesian information criterion (BIC); Hannan–Quinn criterion (HQC). For ADF, lag-order is selected using BIC. For ARCH, lag-order is selected using the partial autocorrelation function (PACF) of squared-inflation. Standard errors of parameters are in parentheses. ** and *** are significance at the 5 % and 1 % levels, respectively. Best specifications are shown by bold numbers.

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/snde-2023-0042).

Received: 2023-06-03
Accepted: 2023-12-07
Published Online: 2024-01-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Multiscale SUR Estimation of Systematic Risk
  4. A Simulation and Empirical Study of the Maximum Likelihood Estimator for Stochastic Volatility Jump-Diffusion Models
  5. Core Inflation Rate for China and the ASEAN-10 Countries: Smoothed Signal for Score-Driven Local Level Plus Scale Models
  6. Diversified Reward-Risk Parity in Portfolio Construction
  7. Time-Varying Parameter Four-Equation DSGE Model
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https://doi.org/10.1515/snde-2023-0042
Keywords for this article
core inflation rate; minimum mean-squared error (MSE) signal extraction; generalized autoregressive score (GAS); dynamic conditional score (DCS); monetary policy

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Multiscale SUR Estimation of Systematic Risk
  4. A Simulation and Empirical Study of the Maximum Likelihood Estimator for Stochastic Volatility Jump-Diffusion Models
  5. Core Inflation Rate for China and the ASEAN-10 Countries: Smoothed Signal for Score-Driven Local Level Plus Scale Models
  6. Diversified Reward-Risk Parity in Portfolio Construction
  7. Time-Varying Parameter Four-Equation DSGE Model
  8. Does State Dependence Matter in Relation to Oil Price Shocks on Global Economic Conditions?
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