--- title: "Introduction to poobly" author: "Christos Adam" output: rmarkdown::html_vignette: number_sections: false word_document: default pdf_document: default fontsize: 11pt urlcolor: blue linkcolor: blue link-citations: true header-includes: \usepackage{float} vignette: > %\VignetteIndexEntry{Introduction to poobly} %\VignetteEncoding{UTF-8} %\VignetteEngine{knitr::rmarkdown} --- ```{r setup, include=FALSE} knitr::opts_chunk$set(echo = TRUE, eval=FALSE) ``` # **Hsiao Poolability test ([1986](#ref-hsiao1986);[2022](#ref-hsiao2022)) in R** ## **Hsiao Poolability test** The Hsiao poolability/homogeneity test (Hsiao [1986](#ref-hsiao1986); [2022](#ref-hsiao2022)) for panel data is used to determine the homogeneity of coefficients across individuals in panel data. The test is composed of three consecutive hypotheses. The hypothesis that slope and intercept (constant) coefficients are the same across the panel is initially tested. If this hypothesis is not rejected, then the total homogeneity (pool) of both slopes and intercepts is concluded. If the first hypothesis is rejected, the second hypothesis tests the homogeneity of the slope coefficients. If this second hypothesis is rejected, different slope coefficients for all individuals are indicated, suggesting total heterogeneity. If this hypothesis is not rejected, then homogeneity of slope coefficients is implied. If this second hypothesis is not rejected, then the the third hypothesis tests the homogeneity of the intercept coefficients across individuals is performed. If this last hypothesis is not rejected, then total homogeneity is resulted. If this last hypothesis is rejected, equal slopes but different intercepts across individuals are indicated. The current implementation is derived from Hsiao ([2022](#ref-hsiao2022)). This procedure is capitulated in Figure [1](#ref-Figure1) and Table [1](#ref-table1).

Fig. 1: Hsiao homogeneity hypothesis testing flow chart.

Fig. 1: Hsiao homogeneity hypothesis testing flow chart.

 Hypothesis Null Alternative
 H1 Pooled H2
 H2 H3 Heterogeneous intercepts & slopes
 H3 Pooled Heterogeneous intercepts & Homogeneous slopes

Table 1: Hsiao homogeneity hypothesis testing table.

## **Example** For this example, the data `Gasoline` from plm R package will be used. ``` r # Import poobly and plm packages to workspace library(poobly) library(plm) # Import "Gasoline" dataset data("Gasoline", package = "plm") # Print first 6 rows head(Gasoline) ``` ``` ## country year lgaspcar lincomep lrpmg lcarpcap ## 1 AUSTRIA 1960 4.173244 -6.474277 -0.3345476 -9.766840 ## 2 AUSTRIA 1961 4.100989 -6.426006 -0.3513276 -9.608622 ## 3 AUSTRIA 1962 4.073177 -6.407308 -0.3795177 -9.457257 ## 4 AUSTRIA 1963 4.059509 -6.370679 -0.4142514 -9.343155 ## 5 AUSTRIA 1964 4.037689 -6.322247 -0.4453354 -9.237739 ## 6 AUSTRIA 1965 4.033983 -6.294668 -0.4970607 -9.123903 ``` A `pdata.frame` or a `data.frame` object is expected as and a formula are required as minimum essential input for the `hsiao` function. Note that `data.frame` object input should be able to be transformed as `pdata.frame` object and `index` input can be used as well. For more about `pdata.frame` see at `plm::pdata.frame`. ```r # Hsiao hypothesis testing x <- hsiao(lgaspcar ~ lincomep + lrpmg + lcarpcap, Gasoline) print(x) ``` ``` ## ## Hsiao Homogeneity Test ## ## Hypothesis| Null | Alternative ## ----------+------+--------------------------------------------- ## H1 |Pooled| H2 ## H2 | H3 | Heterogeneous intercepts & slopes ## H3 |Pooled|Heterogeneous intercepts & homogeneous slopes ## =============================================================== ## ## formula: lgaspcar ~ lincomep + lrpmg + lcarpcap ## ## Hypothesis F-statistic df1 df2 p-value ## 1 H1 129.3166 68 270 < 0.001 ## 2 H2 27.3352 51 270 < 0.001 ## 3 H3 83.9608 17 321 < 0.001 ``` According to this result, the coefficients of the given countries have both heterogeneous intercept & slope. In detail, the first hypothesis, $H_1$, is rejected in 1% statistical significance level, indicating strong evidence against the $H_1$. Then, the second hypothesis, $H_2$, is tested and rejected as well in 1% statistical significance level. This is the end of the testing and so there is strong evidence that both intercept and slope are heterogeneous. ## **References** Hsiao, C. (1986) Analysis of Panel Data. 1st edn. Cambridge: Cambridge University Press (Econometric Society Monographs).
Hsiao, C. (2022) Analysis of Panel Data. 4th edn. Cambridge: Cambridge University Press (Econometric Society Monographs), pp. 43-49. doi:10.1017/9781009057745