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Pesaran, Shin, and Smith Bounds F-Test

Source: R/tests.R
pssf.Rd

This function performs the Pesaran, Shin, and Smith (PSS) F Bound test to assess the presence of a long-term relationship (cointegration) between variables in the context of an autoregressive distributed lag (ARDL) model. The PSS F Bound test examines the joint significance of lagged levels of the variables in the model. It provides critical values for both the upper and lower bounds, which help determine whether the variables are cointegrated. If the calculated F-statistic falls outside these bounds, it indicates the existence of a long-term equilibrium relationship. This test is particularly useful when the underlying data includes a mix of stationary and non-stationary variables.

Usage

pssf(kmodel, case = 3, signif_level = "auto")

Arguments

kmodel

A fitted KARDL model object of class 'kardl_lm' created using the kardl function.

case

Numeric or character. Specifies the case of the test to be used in the function. Acceptable values are 1, 2, 3, 4, 5, and "auto". If "auto" is chosen, the function determines the case automatically based on the model's characteristics. Invalid values will result in an error.

  • 1: No intercept and no trend

  • 2: Restricted intercept and no trend

  • 3: Unrestricted intercept and no trend

  • 4: Unrestricted intercept and restricted trend

  • 5: Unrestricted intercept and unrestricted trend

signif_level

Character or numeric. Specifies the significance level to be used in the function. Acceptable values are "auto", "0.10", "0.1", "0.05", "0.025", and "0.01". If a numeric value is provided, it will be converted to a character string.

When "auto" is selected, the function determines the significance level sequentially, starting from the most stringent level ("0.01") and proceeding to "0.025", "0.05", and "0.10" until a suitable level is identified. Invalid values will result in an error.

Value

A list with class "htest" containing the following components:

  • statistic: The calculated F-statistic for the test.

  • caseTxt: A character string describing the case used for the test, based on the specified case parameter.

  • alternative: A character string describing the alternative hypothesis of the test.

  • sample.size: The number of observations used in the test.

  • varnames: A character vector containing the names of the dependent variable and independent variables used in the test.

  • k: The number of independent variables (excluding the dependent variable) included in the test.

  • sig: The significance level used for the test, either specified by the user or determined automatically.

  • notes: A character vector containing any notes or warnings related to the test, such as the suitability of the test for small sample sizes or any adjustments made to the case based on the model's characteristics.

Hypothesis testing

The null hypothesis (H0) of the F Bound test is that there is no cointegration among the variables in the model. In other words, it tests whether the long-term relationship between the variables is statistically significant. If the calculated F-statistic exceeds the upper critical value, we reject the null hypothesis and conclude that there is cointegration. Conversely, if the F-statistic falls below the lower critical value, we fail to reject the null hypothesis, indicating no evidence of cointegration. If the F-statistic lies between the two critical values, the result is inconclusive.

$$ \Delta {y}_t = \psi + \varphi t + \eta _0 {y}_{t-1} + \sum_{i=1}^{k} { \eta _i {x}_{i,t-1} } + \sum_{j=1}^{p} { \gamma_{j} \Delta {y}_{t-j} }+ \sum_{i=1}^{k} {\sum_{j=0}^{q_i} { \beta_{ij} \Delta {x}_{i,t-j} } }+ e_t $$

Cases 1, 3, 5:

$$\mathbf{H_{0}:} \eta_0 = \eta_1 = \dots = \eta_k = 0$$ $$\mathbf{H_{1}:} \eta_{0} \neq \eta_{1} \neq \dots \neq \eta_{k} \neq 0$$

Case 2:

$$\mathbf{H_{0}:} \eta_0 = \eta_1 = \dots = \eta_k = \psi = 0$$ $$\mathbf{H_{1}:} \eta_{0} \neq \eta_{1} \neq \dots \neq \eta_{k} \neq \psi \neq 0$$

Case 4:

$$\mathbf{H_{0}:} \eta_0 = \eta_1 = \dots = \eta_k = \varphi = 0$$ $$\mathbf{H_{1}:} \eta_{0} \neq \eta_{1} \neq \dots \neq \eta_{k} \neq \varphi \neq 0$$

References

Pesaran, M. H., Shin, Y. and Smith, R. (2001), "Bounds Testing Approaches to the Analysis of Level Relationship", Journal of Applied Econometrics, 16(3), 289-326.

See also

psst ecm narayan

Examples

kardl_model<-kardl(imf_example_data,
                   CPI~ER+PPI+asym(ER)+deterministic(covid)+trend,
                   mode=c(1,2,3,0))
my_pssF<-pssf(kardl_model)
# Getting the results of the test.
my_pssF
#> 
#> 	Pesaran-Shin-Smith (PSS) Bounds F-test for cointegration
#> 
#> data:  model
#> F = 10.204
#> alternative hypothesis: Cointegrating relationship exists
#> 
# Getting details of the test.
my_summary<-summary(my_pssF)
my_summary
#> Pesaran-Shin-Smith (PSS) Bounds F-test for cointegration 
#> F  =  10.20356 
#> k =  3 
#> 
#> Hypotheses:
#> H0: Coef(L1.CPI) = Coef(L1.ER_POS) = Coef(L1.ER_NEG) = Coef(L1.PPI) = 0 
#> H1: Coef(L1.CPI) ≠ Coef(L1.ER_POS) ≠ Coef(L1.ER_NEG) ≠ Coef(L1.PPI)≠ 0 
#> 
#> Test Decision:  Reject H0 → Cointegration (at 1% level) 
#> 
#> Critical Values (Case  V ):
#>          L    U
#> 0.10  3.47 4.45
#> 0.05  4.01 5.07
#> 0.025 4.52 5.62
#> 0.01  5.17 6.36
#> 
#> Notes:
#>    • Trend detected in the model. Case automatically adjusted to 5 (unrestricted intercept and trend).
#> 

# Getting the critical values of the test.
my_summary$crit_vals
#>          L    U
#> 0.10  3.47 4.45
#> 0.05  4.01 5.07
#> 0.025 4.52 5.62
#> 0.01  5.17 6.36


# Using magrittr :

library(magrittr)
   imf_example_data %>%
   kardl(CPI~ER+PPI+asym(ER)+deterministic(covid)+trend,
                           mode=c(1,2,3,0)) %>% pssf()
#> 
#> 	Pesaran-Shin-Smith (PSS) Bounds F-test for cointegration
#> 
#> data:  model
#> F = 10.204
#> alternative hypothesis: Cointegrating relationship exists
#> 

   # Getting details of the test results using magrittr:
   imf_example_data %>%
   kardl(CPI~ER+PPI+asym(ER)+deterministic(covid)+trend,
                           mode=c(1,2,3,0)) %>%
   pssf() %>% summary()
#> Pesaran-Shin-Smith (PSS) Bounds F-test for cointegration 
#> F  =  10.20356 
#> k =  3 
#> 
#> Hypotheses:
#> H0: Coef(L1.CPI) = Coef(L1.ER_POS) = Coef(L1.ER_NEG) = Coef(L1.PPI) = 0 
#> H1: Coef(L1.CPI) ≠ Coef(L1.ER_POS) ≠ Coef(L1.ER_NEG) ≠ Coef(L1.PPI)≠ 0 
#> 
#> Test Decision:  Reject H0 → Cointegration (at 1% level) 
#> 
#> Critical Values (Case  V ):
#>          L    U
#> 0.10  3.47 4.45
#> 0.05  4.01 5.07
#> 0.025 4.52 5.62
#> 0.01  5.17 6.36
#> 
#> Notes:
#>    • Trend detected in the model. Case automatically adjusted to 5 (unrestricted intercept and trend).
#>