Package 'xtdhcoint'

Coef Method for xtdhcoint Objects

Description

Extracts the test statistics from the Durbin-Hausman test.

Usage

## S3 method for class 'xtdhcoint'
coef(object, ...)

Arguments

object	An object of class "xtdhcoint".
...	Additional arguments (currently ignored).

Value

Named numeric vector of z-statistics and p-values.

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Fisher Effect Panel Dataset

Description

Simulated panel dataset for testing the Fisher effect (cointegration between inflation and nominal interest rates). The data are generated with cointegration and common factors, suitable for demonstrating the Durbin-Hausman panel cointegration tests.

Usage

fisher_panel

Format

A data frame with 1000 observations and 4 variables:

country

Character. Country identifier (Country01 through Country20).

year

Integer. Year (1970-2019).

inflation

Numeric. Annual inflation rate (percent).

interest

Numeric. Nominal interest rate (percent).

Details

The data are simulated according to:

$inflation_{it} = \alpha_i + \beta_i \cdot interest_{it} + F_t' \lambda_i + e_{it}$

where $F_t$ are common factors, $\lambda_i$ are factor loadings, and $e_{it}$ is a stationary cointegration error (under cointegration).

The cointegrating coefficient $\beta_i$ is approximately 1, consistent with the Fisher hypothesis.

Source

Simulated data.

Examples

data(fisher_panel)
head(fisher_panel)
table(fisher_panel$country)

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Print Method for xtdhcoint Objects

Description

Prints the results of the Durbin-Hausman panel cointegration test.

Usage

## S3 method for class 'xtdhcoint'
print(x, ...)

Arguments

x	An object of class "xtdhcoint".
...	Additional arguments (currently ignored).

Value

Invisibly returns the input object.

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Summary Method for xtdhcoint Objects

Description

Provides a detailed summary of Durbin-Hausman test results with decision tables at various significance levels.

Usage

## S3 method for class 'xtdhcoint'
summary(object, level = 0.95, ...)

Arguments

object	An object of class "xtdhcoint".
level	Numeric. Confidence level for hypothesis testing. Default is 0.95.
...	Additional arguments (currently ignored).

Value

Invisibly returns a list with summary statistics.

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Durbin-Hausman Panel Cointegration Tests

Description

Performs the Durbin-Hausman panel cointegration tests of Westerlund (2008). The tests are robust to cross-sectional dependence through common factor extraction using principal components.

Usage

xtdhcoint(
  formula,
  data,
  id,
  time,
  kmax = 5,
  criterion = c("ic", "pc", "aic", "bic"),
  penalty = NULL,
  bandwidth = -1,
  predet = 0,
  nfactors = NULL
)

Arguments

formula	A formula specifying the cointegrating relationship (e.g., y ~ x1 + x2). The intercept is never estimated: the unit-specific constant of the levels model vanishes when the data are first-differenced (Westerlund, 2008, eq. 12).
data	A data frame containing panel data. Missing values in the response, the regressors, id or time are not allowed; the function stops and reports their count and row positions.
id	Character string naming the cross-sectional unit identifier.
time	Character string naming the time period identifier. Units are aligned on this variable (see Details, "Unbalanced panels").
kmax	Integer. Maximum number of common factors to consider in the search $k = 0, 1, \ldots, kmax$. Default is 5. Values of at least $\min(N, T - 1)$ are reduced to $\min(N, T - 1) - 1$ with a message.
criterion	Character string specifying the information criterion for factor number selection. One of "ic" (default, the IC criteria of Bai and Ng, 2002, eq. 9), "pc" (their PC criteria), "aic" (their AIC3) or "bic" (their BIC3).
penalty	Integer 1, 2 or 3 selecting the penalty function $g(N, T)$ of the "ic" and "pc" criteria (the default, NULL, means 1). Bai and Ng (2002) define a single AIC3 and a single BIC3, so penalty must be NULL when criterion is "aic" or "bic"; any other combination is an error.
bandwidth	Integer. Bartlett kernel bandwidth $M$ for the long-run variance estimator. If -1 (default) the rule of Westerlund (2008, Section 3.4) is used: $M = \lfloor 4 (T/100)^{2/9} \rfloor$, the same for every unit, where $T$ is the number of time periods in the analysed window. This is a fixed function of $T$, not the data-dependent Newey and West (1994) procedure. A value of 0 uses the contemporaneous variance only. Values other than -1 or a non-negative integer smaller than $T - 1$ are an error.
predet	Numeric. If non-zero, the slope(s) of the first-difference regression are not estimated but set to the given value(s) (Westerlund, 2008, Remark 4). Either a single number applied to every regressor or a vector of length $K$. Default is 0 (estimate the slopes by OLS).
nfactors	Integer or NULL. If supplied, the number of common factors is fixed at this value (0 means no defactoring) and the information criterion is not used. Useful when N is small (see Details).

Details

The Durbin-Hausman tests examine the null hypothesis of no cointegration against the alternative of cointegration. The tests are based on comparing OLS and instrumental variable estimators of the autoregressive parameter in the cumulated residuals.

Computation. Following Westerlund (2008, Section 3.2), for each unit the first differences of the response are regressed on the first differences of the regressors over $t = 2, \ldots, T$ without an intercept (eq. 12). Common factors are extracted from the $(T - 1) \times N$ matrix of residuals by principal components, the defactored differences are recumulated from zero, $\tilde e_{it} = \sum_{j = 2}^{t} \Delta \tilde e_{ij}$ with $\tilde e_{i1} = 0$, and the autoregression (13) is fitted over $t = 2, \ldots, T$. The statistics are therefore invariant to adding a constant to the response or to any regressor.

DHg (Group-mean statistic): Tests the null of no cointegration against the heterogeneous alternative that at least some units are cointegrated. Under the null, the standardized statistic is asymptotically N(0,1).

DHp (Panel statistic): Tests the null of no cointegration against the homogeneous alternative that all units are cointegrated with a common autoregressive parameter. Under the null, the standardized statistic is asymptotically N(0,1).

Both statistics are computed using right-tail critical values. Large positive values indicate evidence of cointegration.

Number of factors. The number of common factors is selected by minimising the chosen criterion over $k = 0, 1, \ldots, kmax$ (Westerlund, 2008, Remark 5; Bai and Ng, 2002, Theorem 2). $k = 0$ means that no factor is removed. For the "pc", "aic" and "bic" criteria the scale $\hat\sigma^2$ is $V(kmax)$, as allowed by Bai and Ng (2002, p. 201).

Number of regressors. The limiting distributions in Westerlund (2008, Theorem 1 and Remark 3) do not depend on the regressors, so the simulated moments of Remark 2 (5.5464, 36.7673, 0.5005, 0.3348) apply for any number of regressors $K$; there is a single set of critical values. What the theory does require is that the regressors are not cointegrated among themselves (Assumption 1(d)). With several regressors the user should check this condition; the function only reports $K$. With predet and $K > 1$, a single value is applied to every regressor; supply a vector of length $K$ for different slopes.

Unbalanced panels. The test is defined for a balanced panel: the common factors $F_t$ are extracted from a $(T - 1) \times N$ matrix whose rows are calendar periods shared by all units (Westerlund, 2008, Section 3.1 and p. 201), and the criterion of Remark 5 sums over $t = 2, \ldots, T$ for every unit. When units are observed over different periods the function keeps the common time window, the set of values of time at which every unit is observed, drops the observations outside it with a message, and requires that window to consist of consecutive periods of the data (no unit may have a gap inside it). If the common window has gaps, or fewer than 10 periods, the function stops with an error. Duplicate values of time within a unit are an error.

Small N and N larger than T. The limiting distributions of Theorem 1 require both $N$ and $T$ to grow, with $N/T$ going to zero. In a Monte Carlo study of this version (Westerlund's Section 3.4 design, nominal 5 percent level, one factor fixed with nfactors = 1, 1000 replications for each of two seed sets) the null rejection rate was 38 to 41 percent for DHp with $N = 2$ ($T = 50$ and $100$), 23 to 24 percent for DHp with $N = 5$ ($T = 50$ and $100$), and 57 to 60 percent for DHg with $N = 25$ and $T = 20$. The function therefore warns when $N < 5$ or $N > T$. The Bai and Ng (2002) criteria are also asymptotic in both $N$ and $T$: "for very small N or T, the proposed criteria are inadequate" (p. 203). In their Table I ($r = 1$, $k_{max} = 8$) with $N = 10$, every criterion except BIC3 averages the upper bound 8, whereas BIC3 averages 7.18 ($T = 50$) and 5.88 ($T = 100$). The function warns when $N < 10$ and the number of factors is selected automatically; fix the number of factors with nfactors or report results for several values of kmax.

Value

An object of class "xtdhcoint" containing:

dhg

Raw DHg (group-mean) statistic.

dhp

Raw DHp (panel) statistic.

dhg_z

Standardized DHg z-statistic.

dhp_z

Standardized DHp z-statistic.

dhg_pval

P-value for DHg (right-tail).

dhp_pval

P-value for DHp (right-tail).

nf

Number of common factors used (estimated, or fixed by nfactors). 0 means that no factor was removed.

nf_at_kmax

Logical. TRUE when the information criterion selected the upper bound kmax.

nfactors

The value of the nfactors argument.

N

Number of cross-sectional units.

TT

Number of time periods in the analysed window.

time_window

The first and last value of time in the analysed window.

n_dropped

Number of observations outside the common time window that were dropped.

K

Number of regressors.

bandwidth

Bandwidth used.

kmax

Upper bound of the factor search actually used.

criterion

Information criterion used.

penalty

Penalty function used (NA for "aic" and "bic").

call

The matched call.

References

Westerlund, J. (2008). Panel cointegration tests of the Fisher effect. Journal of Applied Econometrics, 23(2), 193–233. doi:10.1002/jae.967

Bai, J. and Ng, S. (2002). Determining the number of factors in approximate factor models. Econometrica, 70(1), 191–221. doi:10.1111/1468-0262.00273

Newey, W. K. and West, K. D. (1994). Automatic lag selection in covariance matrix estimation. Review of Economic Studies, 61(4), 631–653. doi:10.2307/2297912

Examples

# Load example data
data(fisher_panel)


# Basic test with default settings
result <- xtdhcoint(inflation ~ interest, data = fisher_panel,
                    id = "country", time = "year")
print(result)
summary(result)

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