Inference for Income Mobility Measures in the Presence of Spatial Dependence

Abstract

Income mobility measures provide convenient and concise ways to reveal the dynamic nature of regional income distributions. Statistical inference about these measures is important especially when it comes to a comparison of two regional income systems. Although the analytical sampling distributions of relevant estimators and test statistics have been asymptotically derived, their properties in small sample settings and in the presence of contemporaneous spatial dependence within a regional income system are underexplored. We approach these issues via a series of Monte Carlo experiments that require the proposal of a novel data generating process capable of generating spatially dependent time series given a transition probability matrix and a specified level of spatial dependence. Results suggest that when sample size is small, the mobility estimator is biased while spatial dependence inflates its asymptotic variance, raising the Type I error rate for a one-sample test. For the two-sample test of the difference in mobility between two regional economic systems, the size tends to become increasingly upward biased with stronger spatial dependence in either income system, which indicates that conclusions about differences in mobility between two different regional systems need to be drawn with caution as the presence of spatial dependence can lead to false positives. In light of this, we suggest adjustments for the critical values of relevant test statistics.

Keywords

income mobility spatial dependence statistical inference Monte Carlo Markov chains

Income inequality is an important subject of interest around the world. Many indices intended for measuring the income inequality of an economic system at a given time point have been developed and popularized, including, but not limited to, the Gini index, coefficient of variation, and Theil’s measure (Allison 1978; Shorrocks 1980). However, concern is not only about individuals’/households’ current economic status but also where they would end up and their lifetime welfare (Creedy and Wilhelm 2002; Ruiz-Castillo 2004; Khor and Pencavel 2008). It is becoming increasingly recognized that a static view of the income distributions cannot reveal the whole picture and that the dynamics of income distribution shapes social welfare as well (Shorrocks 1978a; Chakravarty 1995; Maasoumi 1998). Thus, income mobility measures, which evaluate the changes in economic status over time or generations, serve as a complement to income inequality measures to reveal a fuller picture of income inequality dynamics and social welfare (Fields and Ok 1996, 1999).

Similar issues arise when the focus shifts from the distribution of incomes taken over individuals/households in a society to the question of income distributions of regions (Rey 2015). That is, in a national system, what are the properties of the distribution of regional incomes, and how do these evolve over time? Similarly, regional income mobility measures offer a concise way to reveal the dynamic nature of the regional income distribution and serves as a complement to regional income inequality measures. There are two main types of income mobility: structural mobility and exchange mobility (Ruiz-Castillo 2004). The former measures absolute income changes over time, while the latter measures income changes relative to one another. When one is silent in some cases, the other might be able to identify some important mobility patterns. For example, if all the regions encounter the same level of economic growth, their income rank positions remain unchanged. In this case, the exchange mobility measures would not pick up anything, while the structural mobility measures could. On the other hand, if the regions only exchange income values, the structural mobility measures would be silent, while the exchange mobility measures would not. Thus, these two types serve as complements to one another.

Statistical inference about regional income mobility measures is of great importance if a confidence interval is to be constructed for the estimate (Schluter 1998), let alone when it comes to a comparison of two regional systems. Rey and Ye (2010) compared the regional income mobility over 1978–1998 between the United States and China based on permutation-based sampling distributions. A theoretical inference framework has been built in Trede (1999) assuming regional time series are independently and identically distributed. However, spatial effects including spatial dependence and spatial heterogeneity are known as more of a rule than exception in a regional context, which poses a serious question: would the spatial effects impair classic inference so significantly that they could not be ignored? This question motivates our research. Here, we focus on the so-called Markov-based mobility measures. We expect to expose the nature of the impact of spatial dependence on the inference through a series of Monte Carlo simulation experiments. To do this, we propose a novel data generating process (DGP) capable of generating spatially dependent Markov chains given a transition probability matrix and a specified level of spatial dependence. Results suggest that spatial dependence does have a major influence on the properties of the mobility estimators and relevant test statistics. Although it does not bias the maximum likelihood estimators (MLEs) of the mobility measures, it dramatically increases the variances of their sampling distributions, raising the Type I error rate for one-sample tests. As for the two-sample tests, the size tends to become increasingly upward biased with stronger spatial dependence in either income system while the power decreases with stronger spatial dependence. The asymptotic properties originating from MLEs do not hold for small sample sizes: not only the variance is underestimated, but also the MLEs are biased.

For the rest of the article, we first introduce the definition of three mobility measures, as well as the respective estimators, one-sample and two-sample test statistics. Then, a novel DGP for producing spatially dependent Markov chains is proposed and adopted in a series of Monte Carlo simulation experiments intended for examining the properties of the aforementioned mobility estimators and test statistics. Next, we discuss the experimental results and propose adjustments to the critical values of the tests to maintain proper size and power properties. In the final section, we conclude and suggest some further research directions.

Regional Income Mobility Measures

In this article, we focus on Markov-based mobility measures. The motivation is that discrete Markov chain (DMC) theory has been widely applied in studying regional income dynamics and convergence (e.g., Quah 1996; Le Gallo and Chasco 2008; Liao and Wei 2012; Rey and Sastré Gutiérrez 2015) since the estimated transition probability matrix P can reveal abundant information on transition probabilities across discretized income states over time. However, the matrix P , comprised of m² elements (m is the number of discrete states adopted to discretize the income distribution), is not as simple and straightforward as a single index especially when it comes to comparing two regional income systems. In this context, several Markov-based mobility measures have been proposed in the literature, all of which can be calculated from the estimated transition probability matrix.¹ Thus, we start by briefly introducing DMC theory and then proceed to derive relevant mobility measures.

DMCs

As mentioned before, the transition probability matrix P , which is the core of DMC, contains information regarding mobility across discrete states over time. Equation (1) displays an example of such matrix in which $p_{i j}$ represents the probability of transitioning from state i to state $j$ over a given time interval.

P = [\begin{matrix} p_{11} & p_{12} & \dots & p_{1 m} \\ p_{21} & p_{22} & \dots & p_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ p_{m 1} & p_{m 2} & \dots & p_{m m} \end{matrix}],0 \leq p_{i j} \leq 1, \sum_{j = 1}^{m} p_{i j} = 1 \forall i, j \in S = {1, 2, \dots, m} .

Here m states are adopted to discretize the income data. Class boundaries, as well as preliminary transformations of incomes, are determined by the user. Caution should be taken when making such decisions as different strategies might lead to different results and conclusions regarding income dynamics. For further discussion on the issue, please refer to Rey (2015).

Each row of P could be viewed as a multinomial distribution conditioned on the preceding state. For example, the second row of P represents the respective probabilities of transitioning to each of the m states at t, given that an observation was in the second state at $t - 1$ . Since these multinomial distributions are conditionally independent, the MLE for each individual transitional probability could be derived as shown in equation (2), where $n_{i j}$ is the number of transitions from state i to state $j$ (Anderson and Goodman 1957). Usually, a single transition probability matrix is estimated from the pooled income data across space and time. For the matrix to hold as the “ubiquitous” dynamic rule, several assumptions must be valid. Shorrocks (1976) presents three major assumptions:

First-order Markov: the income dynamic system has such a short memory that its current state is only influenced by the immediate past.

Population homogeneity: the same transition probabilities apply to all regions being studied.

Time homogeneity: the transition probabilities remain constant over time.

{\hat{p}}_{i j} = \frac{n_{i j}}{\sum_{q = 1}^{m} n_{i q}} .

However, meticulous inspection of the above assumptions reveals its potential defect for applications in regional contexts. If there exists cross-sectional spatial dependence (Rey, Kang, and Wolf 2016), which is very much likely, the assumption of random sampling that underlies the properties of the MLEs of the transition probabilities will be violated. As such, the properties of these estimators and any mobility measure derived from them may be impaired.

Mobility Measures

A continuous real function $M (\cdot)$ is defined over the set of transition probability matrices to produce a real-value mobility measure. We concentrate on the following three mobility measures:

M_{1} (P) = \frac{m - \sum_{i = 1}^{m} p_{i i}}{m - 1},

M_{2} (P) = 1 - | d e t (P) |,

M_{3} (P) = 1 - | λ_{2} |,

where $d e t (P)$ is the determinant of P and $λ_{i}$ represents the eigenvalue of P and $1 = | λ_{1} | > | λ_{2} | > \dots > | λ_{m} |$ . $M_{1}$ can be considered as the probability of leaving a class. As demonstrated in Prais (1955), the expected length of stay in class i is $\frac{1}{1 - p_{i i}}$ . Normalizing the reciprocal of the harmonic mean of the expected length of stay for every class by $\frac{m}{m - 1}$ produces $M_{1}$ (Shorrocks 1978b). $M_{2}$ is the difference between $1$ and the absolute value of the determinant of the transition probability matrix (Shorrocks 1978b). The final measure utilizes the absolute value of the second largest eigenvalue and deducts it from $1$ (Sommers and Conlisk 1979). Based on spectral theory, the largest eigenvalue of P is $1$ ( $λ_{1} = 1$ ), and the remaining ones have absolute values less than $1$ . What is relevant here is that the absolute value of the second largest eigenvalue $λ_{2}$ determines the rate of the convergence of the Markov chain. That is, the smaller the ${|λ}_{2} |$ is, the faster the chain converges. We will refer to these three mobility measures as Shorrocks mobility, determinant mobility, and eigenvalue mobility, respectively, for the rest of the article.

For any transition probability matrix with a quasi-maximal diagonal, all of the three mobility measures take values on $[0, 1]$ .² $0$ refers to immobility and $1$ perfect mobility. Intuitively, if the transition probability matrix takes the form of the identity matrix, every region is stuck in its current state implying complete immobility. On the contrary, when each row of P is identical, current state is irrelevant to the probability of moving away to any class. Thus, the transition matrix with identical rows is considered perfect mobile. Although all three mobility measures have the same bounds, we should not expect that they are comparable to each other. As we shall see later, the mean and variance of these measures are rather different.

Another important property of mobility measures is monotonicity. Suppose that we increase one off-diagonal element at the expense of the diagonal element in the same row, we would expect the mobility measure to be able to pick up the change by raising its value. We will utilize this property in designing the Monte Carlo experiments.

Statistical Inference

Mobility estimator

The natural estimators for the three mobility measures are $M_{1} (\hat{P})$ , $M_{2} (\hat{P})$ , and $M_{3} (\hat{P})$ , where $\hat{P}$ is an MLE whose elements are defined in equation (2). Asymptotically, $\hat{P}$ follows a multivariate normal distribution with the variance–covariance matrix $Σ_{\hat{P}}$ defined in equation (6):

c o v ({\hat{p}}_{i j}, {\hat{p}}_{k l}) = {\begin{array}{l} \frac{{\hat{p}}_{i j} (1 - {\hat{p}}_{i j})}{n {\hat{π}}_{i}} & if i = k and j = l, \\ - \frac{{\hat{p}}_{i j} {\hat{p}}_{i l}}{n {\hat{π}}_{i}} & if i = k and j \neq l, \\ 0 & else, \end{array}

where n is the total number of transitions and $\hat{π}$ is the estimator for the steady-state distribution ( $\hat{π} P = \hat{π}$ ). To derive the asymptotic variance for mobility measures, the delta method could be utilized. Let $M (\hat{P})$ represents any of the three measures. Then, the estimator of the asymptotic variance for $M (\hat{P})$ is as follows:

σ_{M (\hat{P})}^{2} = d Σ_{\hat{P}} d',

where d is the $(1, m^{2})$ derivative vector of $M (P)$ with respect to P as shown in equation (8) and $d'$ is the transpose of d .

d = v e c ((\frac{\partial M (P)}{\partial (P)})')' .

Here, $v e c$ converts a matrix into a column vector by stacking the columns on top of one another. For the three mobility measures we consider, the derivatives are obtained as follows (Trede 1999):

d_{M_{1}} = - \frac{1}{m - 1} v e c (I)',

d_{M_{2}} = - s i g n (d e t (P)) v e c (\tilde{P}')',

d_{M_{3}} = - v e c ({\overset{⌣}{P}'}_{λ_{2}})',

where I is the $(m, m)$ identity matrix, $\tilde{P}$ is the cofactor matrix of P , $s i g n (d e t (P))$ extracts the sign of $d e t (P)$ , and ${\overset{⌣}{P}}_{λ_{2}}$ is the derivative of the second absolute largest eigenvalue with respect to P .

With these derivatives in hand, we are able to calculate the asymptotic variance of $M (\hat{P})$ . As shown in Trede (1999), the asymptotic sampling distribution of the estimator for each of the above three mobility measures follows a normal distribution with mean $M (\hat{P})$ and variance $σ_{M (\hat{P})}^{2}$ . We are going to investigate how the contemporaneous spatial dependence across regional income time series impacts the properties of each of the three estimators.

One-sample test

It might be the case that we want to know whether the economic mobility of a regional system is equal to or lower/higer than a specific level. A one-sample test about the mobility measure could serve the purpose as shown in equation (12):

z_{1} = \frac{M - x}{σ_{M}},

where $z_{1}$ is the test statistic, M is the observed mobility estimate (for Shorrocks mobility, determinant mobility, or eigenvalue mobility), $x$ is a value between $0$ and $1$ representing the anticipated mobility level we want to test against, and $σ_{M}$ is the analytical standard deviation of M. Because M is asymptotically normally distributed, $z_{1}$ obeys the standard normal distribution asymptotically under the null hypothesis $H_{0} : M = x$ .

Two-sample test

For a mobility comparison of two income systems, such as the United States (system A) and China (system B), a two-sample test is required. Since it is known that the asymptotic sampling distribution of the estimator is a normal distribution, a two-sample z-test can be utilized to serve the purpose. The test statistic is defined in equation (13):

z_{2} = \frac{M^{(A)} - M^{(B)}}{\sqrt{σ_{M^{(A)}}^{2} + σ_{M^{(B)}}^{2}}},

where $M^{(A)}$ and $M^{(B)}$ are mobility measures estimated from income dynamic systems A and B based on the same mobility function, such as $M_{1}$ , $M_{2}$ , or $M_{3}$ . The null hypothesis is $H_{0} : M^{(A)} = M^{(B)}$ while three alternatives can be specified as $H_{a} : M^{(A)} \neq M^{(B)}$ , $H_{a 1} : M^{(A)} > M^{(B)}$ , and $H_{a 2} : M^{(A)} < M^{(B)}$ , leading to the two-tail test, upper-tail test, and lower-tail test. Under each null, the asymptotic sampling distribution of the test statistic is the standard normal distribution, that is, $z_{2} \sim N (0, 1)$ .

Various factors might impact the properties of this test statistic as it concerns two systems. Interaction between two income systems is one potential cause, though we are not going to investigate it in this article. We will always assume that the two systems being compared are independent of one another. Another factor concerns the discretization strategy. Application of identical classification boundaries to the real income values of the two systems appears to be the natural way to proceed, but the possible unequal development status (e.g., the United States and China) will lead to an almost absolute rejection of the null. Normalizing the real incomes by the average and then using the quantile discretization strategy would adjust for these differences in development status when testing for differences in mobility between the two systems. Here, the mobility comparison considered is relative mobility rather than absolute mobility.

In addition to these two issues, contemporaneous spatial dependence across regional income time series in either system might impair the properties of the test statistic. We will investigate its impact via a series of Monte Carlo simulation experiments.

Monte Carlo Experiment

In this section, we introduce a series of Monte Carlo simulation experiments that are designed to examine the impact of contemporaneous spatial dependence between regional time series on the properties of mobility measure estimators and relevant test statistics. Here, the spatial dependence we consider is the so-called substantive spatial dependence rather than nuisance spatial dependence (Anselin 1988). The former is part of the underlying process, while the latter is not.

DGP

That all the three mobility measures are derived from the transition probability matrix P makes P the core of our DGP. That is, we need to propose a DGP that generates time series mimicking the Markov chain governed by the transition matrix P . The other significant factor we need to incorporate in the DGP is the contemporaneous spatial dependence between time series. In the following sections, we first introduce a common approach to simulating a Markov chain given P , followed by an extended approach to simulating a set of spatially dependent Markov chains given P and spatial dependence level $ρ$ .

Generating a Markov chain

The most common approach to producing a realization of a first-order Markov chain ${X_{0}, X_{1}, \dots, X_{t}}, t > 0$ utilizes the continuous uniform distribution defined over the range $(0, 1)$ . The cumulative distribution function (CDF) for the uniform distribution is a simple diagonal line $F (x) = x, x \in (0, 1)$ . Starting with a simple two-state Markov chain with the transition probability matrix $P_{p}$ defined in equation (14), we need to transform $P_{p}$ into a cumulative probability matrix (CPM) first. As mentioned before, each row of $P_{p}$ is a multinomial distribution conditional on the preceding state. That is to say, if the region is in state 1 at t, then the probability of transitioning to state 1 and 2 at $t + 1$ are $0.7$ and $0.3$ , respectively. Similarly, if the region is in state 2 at t, the probability of transitioning to state 1 and 2 at $t + 1$ are $0.5$ and $0.5$ . To construct, the CPM is to calculate cumulative probabilities for each row. Thus, the CPM for $P_{p}$ would be $P_{c}$ as shown in equation (14):

P_{p} = \begin{matrix} 1 \\ 2 \end{matrix} \begin{array}{l} \begin{array}{l} 1 2 \\ [\begin{array}{l} 0.7 0.3 \\ 0.5 0.5 \end{array}] \end{array} \end{array}; P_{c} = \begin{matrix} 1 \\ 2 \end{matrix} \begin{array}{l} \begin{array}{l} 1 2 \\ [\begin{array}{l} 0.7 1.0 \\ 0.5 1.0 \end{array}] \end{array} \end{array} .

Suppose we need to simulate a Markov chain with length $t > 3$ given the initial state $X_{0} = 2$ , t random numbers are generated from the continuous uniform distribution. Let’s say they are $u = {0.7, 0.2, 0.8, \dots}$ . Because $X_{0} = 2$ , we pick the second row of $P_{c}$ to determine the state at $t = 1$ . As the cumulative probability of the random number $0.7$ is $0.7$ , which is greater than the cumulative probability of the first state $0.5$ and smaller than that of the second state $1.0$ , we assign 2 to the state at $t = 1$ . The next two states would be determined in a similar fashion. In the end, we would end up with the simulated Markov chain ${2, 2, 1, 2, \dots}$ . With t large enough, the maximum likelihood estimation of the transition matrix would be very similar to the true matrix $P_{p}$ .

The rule for determining the state of $X_{t}$ could be generalized as follows: compare the cumulative probability $c p_{t}$ of the generated random number $u_{t}$ and the cumulative probabilities of all m states conditional on $X_{t - 1}$ . That is to say, if $X_{t - 1} = k, k \in {1, 2, \dots, m}$ , the kth row of the CPM would be utilized. If $c p_{t} < {CPM}_{k 1}$ , assign 1 to $X_{t}$ ; if not, proceed to ${CPM}_{k 2}$ . If $c p_{t} < {CPM}_{k 2}$ , 2 is assigned to $X_{t}$ ; if not, proceed to the next state ${CPM}_{k 3}$ . Since the cumulative probability of the last state is always 1, $X_{t}$ should always be rightfully determined.

To summarize, the procedures of producing a T-long realization of a Markov chain given a initial state $X_{0}$ and a transition probability matrix P are:

Construct the CPM of P .

Generate T random samples (Markov innovations) ${u_{1}, u_{2}, \dots u_{T}}$ from the continuous uniform distribution. Set $j = 1$ .

Use the above determination rule to find the state for $X_{j}$ .

If $j < T$ , repeat step (3), otherwise stop.

In the case of a collection of N regions, we can repeat this process N times to generate N independent Markov chains. If we collect the Markov innovations in the matrix U of size $N \times T$ , we note that each pair of rows $i \neq j$ have pairwise 0 covariance $c o v [U_{i,.}, U_{j,.}] = 0$ . In other words, the innovation for region $j$ in period t is independent of the innovation for region i in the same period.

Generating a set of spatially dependent Markov chains

In the regional setting, we are confronted with a number of time series each of which is the income trajectory of a specific region. Since common practice is to estimate one transition probability matrix P from the pooled data set, the implicit assumption would be that P holds for every region. The complication here is that P would be a ubiquitous dynamic rule indeed, but the estimator (equation [2]) might be impaired if these time series are correlated to some degree. Our interest lies in the influence of potential spatial dependence between time series. Thus, a DGP producing a set of spatially dependent time series each of which is governed by a common given transition probability matrix is required.

Our approach is based on four steps:

Construct the CPM of P .

Draw T samples from an N-dimensional joint normal distribution with a specified level of spatial dependence. Define this as a matrix U with size $N \times T$ .

Derive N marginal univariate CDFs based on which the cumulative probability of each element $i = 1, 2, \dots, N$ in sample $t (t \in [1, T])$ , $u_{i t}$ , could be obtained.

Apply the determination rule to the CPM of P and the cumulative probabilities from the previous step for selecting the next state in the Markov chain currently in state $X_{i t}$ .

For step (b), we employ the spatial lag model (SAR) to produce spatially dependent cross-sectional data:

U_{t} = ρ W U_{t} + ϵ_{t},

where $U_{t}$ is a $(1, N)$ vector of random variates at time t, $ρ \in [0, 1)$ is the level of spatial dependence constant over time, $W$ is the row-normalized spatial weight matrix indicating the interaction between regions, and $ϵ_{t}$ is a vector of random errors independently and identically distributed as a normal distribution $ϵ_{t i} \sim N (μ_{ϵ}, σ_{ϵ}^{2}), i \in {1, 2, ..., N}$ (N is the number of regions). Rewriting equation (15) in reduced form, we acquire:

U_{t} = (1 - ρ W)^{- 1} ϵ_{t} .

Since $ϵ_{t}$ follows a multivariate normal distribution, $U_{t}$ also follows a multivariate normal distribution with a variance–covariance matrix whose nondiagonal elements are not necessarily 0 when $ρ$ is not equal to 0. More specifically,

U_{t} \sim N (μ_{ε}, σ_{ε}^{2} {(I - ρ W)}^{- 1} ((I - ρ W)^{- 1})') .

We then convert the these series to the Markov states based on steps (c) to (d). Note that when $ρ = 0$ , this approach collapses to the case of simulating N independent DMCs as in the previous section, since now the rows of the matrix U are pairwise independent. In contrast, when $ρ \neq 0$ , the N rows of U are no longer independent, and thus, the N Markov chains are spatially correlated.

Simulation Design

A set of simulation experiments that are designed to examine the impact of contemporaneous spatial dependence on the sampling distribution of the (three) estimator(s), as well as the size and power of the (three) test statistic(s), is introduced in this section.

Monotone Markov matrix

As illustrated before, the DGP requires a specification of a transition probability matrix P . We restrict our research to the so-called monotone Markov matrix, which is usually encountered in empirical economic analysis. A transition matrix is considered monotone if each row stochastically dominates the row above it (Conlisk 1990). As a consequence, the probability of any region transitioning to better-off states would be higher next period if it is currently in state $i + 1$ than i. One important implication of the monotone transition matrix is given in Dardanoni (1995) as Lemma 1, which states that if two regions are faced with a common monotone transition probability matrix, the income distribution for region l would always stochastically dominate that for region h if the initial income distribution for region l stochastically dominate that for region h, though both regions would converge to a common steady-state distribution in the long run. This echoes the neoclassical economic growth theory (Barro and Sala-i Martin 2003) in the sense of all regions monotonically converging to a common steady state. A major difference to be noticed here is that the neoclassical economic growth theory describes the income trajectory in a more deterministic sense, while the monotone Markov chain is a stochastic model. Thus, the monotone Markov chain leaves more space for intradistributional dynamics such as leapfrogging.

Experiments for mobility estimator and one-sample test

We adopted a $5 \times 5$ transition probability matrix $P 5$ which was estimated from the discretized (quantiles) relative US state income time series 1929–2010 for the DGP. It is obvious that $P 5$ is a monotone transition matrix:

P 5 = [\begin{matrix} 0.915 & 0.075 & 0.009 & 0.001 & 0.000 \\ 0.066 & 0.827 & 0.105 & 0.001 & 0.001 \\ 0.005 & 0.103 & 0.794 & 0.095 & 0.003 \\ 0.000 & 0.009 & 0.094 & 0.849 & 0.048 \\ 0.000 & 0.000 & 0.000 & 0.062 & 0.938 \end{matrix}] .

In addition to the transition matrix, the DGP requires the specification of sample size $(N, T)$ , a spatial weighting matrix $W$ , a level of spatial dependence $ρ$ , initial states and the parameters $(μ_{ϵ}, σ_{ϵ}^{2})$ of the normal distribution for the error term. To investigate whether the asymptotic properties of the three estimators hold in small sample settings, we incorporated $N = 25, 169$ and $T = 50, 200$ in our simulation experiments. The spatial configuration was a $N^{\frac{1}{2}} \times N^{\frac{1}{2}}$ regular grid based on which a rook contiguity weight matrix is constructed and used in the DGP. We varied spatial dependence levels $ρ = 0, 0.2, 0.5, 0.7, 0.9, 0.98$ to investigate the pattern of impacts imposed by dependence and whether there was a threshold value above which the impact could not be readily ignored. The initial states were randomly assigned and $μ_{ϵ} = 0, σ_{ϵ}^{2} = 0.5$ throughout the experiments.

For each combination of parameters, we simulated the DGP $1, 000$ times and built the empirical sampling distribution for each of the three mobility estimators. Since we knew the “true” transition probability matrix $P 5$ , we could analytically derive the asymptotic sampling distribution under the circumstances of no spatial dependence. Comparing the empirical and analytical asymptotic distributions would shed light on the influence of contemporaneous spatial dependence in small and large sample settings.

Experiments for two-sample test statistic

To investigate the properties of the two-sample test statistic, we need to simulate two dynamic systems which requires two transition probability matrices $P^{(A)}$ and $P^{(B)}$ . $P^{(A)}$ serves as the dynamic rule for system A and $P^{(B)}$ for system B. As the null hypothesis is that both systems share a common mobility value, we used the same transition matrix $P 5$ for both systems. That is, $P^{(A)} = P^{(B)} = P 5$ .

To examine the power of the two-sample tests for three different alternatives $H_{a}$ , $H_{a 1}$ , and $H_{a 2}$ , we need to come up with another transition probability matrix that is different from the baseline matrix $P 5$ . The intuitive approach is to adjust the elements of $P 5$ in a systematic way, so that we have control over the direction and magnitude of the difference in terms of mobility.

As we have mentioned earlier, all of the three mobility measures have an important property, monotonicity. Dardanoni (1995) discussed a type of perturbation to a transition matrix called “diagonalising shift,” which decreases mobility by shifting probability mass toward the main diagonal. Here, we slightly adjust the approach to make it more systematic and operable. Instead of shifting toward the main diagonal, we shift from it. In order to control the magnitude of the shifting, we shift a certain portion $β \in [0, 1)$ at a time. As shown in equation (19), the shifted mass is proportionally assigned to the nondiagonal elements in each row. For example, if we are to investigate the power of the tests when the mobility difference between two systems is small, we can adopt a small portion $β = 0.01$ in the adjusted diagonalizing shift method. Thus, the new transition probability matrix $P 5_{0.01}$ is acquired as shown in equation (20). By assigning $P^{(A)} = P 5$ and $P^{(B)} = P 5_{0.01}$ in the DGP, we could simulate two regional income systems governed by two different transition probability matrices.

\begin{matrix} p_{i i}^{new} = (1 - β) p_{i i}, i \in {1, \dots, m}, \\ p_{i j}^{new} = p_{i j}^{new} + \frac{β p_{i i}}{m - 1}, j \in {1, \dots, m}, j \neq i . \end{matrix}

P 5_{0.01} = [\begin{matrix} 0.906 & 0.077 & 0.011 & 0.003 & 0.002 \\ 0.068 & 0.819 & 0.107 & 0.003 & 0.003 \\ 0.007 & 0.105 & 0.786 & 0.097 & 0.005 \\ 0.002 & 0.011 & 0.096 & 0.841 & 0.050 \\ 0.002 & 0.002 & 0.002 & 0.064 & 0.929 \end{matrix}] .

When $β = 0$ , the new transition probability matrix would be the same as $P 5$ . To examine the power of the two-sample test, we also varied $β = 0.01, 0.03, 0.05$ to investigate the sensitivity of the tests to contemporaneous spatial dependence under different circumstances. The “true” mobility differences for varied $β$ based on the three measures are shown in Table 1. The determinant mobility measure tends to give the largest difference. It is almost twice the difference obtained from the other two measures.

Table 1.

True Mobility Differences.

Mobility Measure	$M^{(A)}$	Difference $M^{(A)} - M^{(B)}$
Mobility Measure	$M^{(A)}$	$β = 0.01$	$β = 0.03$	$β = 0.05$
$M_{1}$	.169	−.011	−.032	−.054
$M_{2}$	.540	−.024	−.068	−.110
$M_{3}$	.041	−.011	−.034	−.057

Besides the two transition matrices, $P^{(A)}$ and $P^{(B)}$ , the other parameters needed for the DGP were the same as that used in the experiments for mobility estimators. That is, $N = 25, 169$ , $T = 50, 200$ , a rook contiguity weight matrix for regular lattice, $ρ^{(A)} = 0, 0.2, 0.5, 0.7, 0.9, 0.98$ , $ρ^{(B)} = 0, 0.2, 0.5, 0.7, 0.9, 0.98$ , $μ_{ϵ} = 0$ , and $σ_{ϵ}^{2} = 0.5$ . For each combination of parameters, we simulated from the DGP $2, 000$ times ( $1, 000$ for $P^{(A)}$ and 1,000 for $P^{(B)}$ ). For each set of simulated data sets, we calculated three test statistics, each for one type of mobility measures, and recorded rejection ratios at the $α = 0.05$ significance level.

Results

Sampling Distributions of Mobility Estimators

We start with looking at the sampling distributions of three mobility estimators $M_{1} (\hat{P})$ , $M_{2} (\hat{P})$ , and $M_{3} (\hat{P})$ . As discussed earlier, when the regional time series are free of spatial dependence, the asymptotic analytical sampling distribution for each measure is a normal distribution with the mean and variance determined by the underlying dynamic rule (the transition probability matrix P ) and the sample size $N, T$ . Since we know the “true” transition probability matrix, we also know the analytical sampling distribution. By comparing it with the empirical sampling distribution constructed from 1,000 simulated samples under various circumstances, we could observe the impact of contemporaneous spatial dependence as well as sample size.

For Shorrocks mobility estimator $M_{1} (\hat{P})$ , Figure 1 shows the asymptotic analytical and empirical sampling distributions. The red curve depicts the former, while the gray curves display the latter. The darker gray curves represent higher levels of spatial dependence. Each subplot represents a different sample size. The subplots in the upper row display the sampling distributions when $T = 50$ , while those in the lower row $T = 200$ . The subplots in the left column display the sampling distributions when $N = 25$ , while those in the right column correspond to $N = 169$ . Thus, the upper-left subplot shows the case when sample size is fairly small $N = 25, T = 50$ , and the lower-right one shows a large sample case $N = 169, T = 200$ .

Figure 1.

Asymptotic analytical and empirical sampling distributions of the shorrocks mobility estimator $M_{1} (\hat{P})$ .

We can observe from the lower-right subplot that when $ρ = 0$ , the empirical distribution fits quite well with the asymptotic analytical distribution. As $ρ$ increases, it is still a normal distribution, but the variance increases dramatically. The robustness of the normality of the distribution to the presence of spatial autocorrelation has been validated by conducting several normality tests including the Kolmogorov–Smirnov test, Shapiro–Wilk test (Shapiro and Wilk 1965), and D’Agostino and Pearson’s (1973) normality test, none of which rejects the null hypothesis of a normal distribution. When spatial dependence is very strong $ρ = 0.98$ , the sampling variance can reach twenty-eight times the analytical variance. On the other hand, the mean doesn’t seem to deviate from the analytical mean until $ρ = 0.98$ .

Moving to the upper-left subplot where sample size is small, the pattern is a little different. Even when $ρ = 0$ , the empirical distribution doesn’t seem to fit well with the asymptotic analytical distribution.³ It is a little more dispersed and slightly shifts to the right of the latter. In other words, the asymptotic properties do not hold for small sample sizes: not only the variance of the estimator is underestimated, but also the mobility estimator is biased. Therefore, the actual significance level would be larger than $0.05$ leading to a higher Type I error rate even the regional economic system is exempt from spatial dependence. When there is spatial dependence between time series at work, both the variance and mean grow dramatically with $ρ$ increasing.

Figures A1 and A2 (see Appendix A) show the asymptotic analytical and empirical sampling distributions for different sample sizes and under varied spatial dependence levels for mobility estimators $M_{2} (\hat{P})$ and $M_{3} (\hat{P})$ . The general pattern is quite similar to Shorrocks mobility estimator $M_{1} (\hat{P})$ . That is, as the spatial dependence strength becomes stronger, the empirical sampling distribution would still stay as a normal distribution though the variance grows dramatically and the mean grows mildly. In addition, the asymptotic properties do not seem to hold well in small sample settings, at least not when $N = 25, T = 50$ .

The dramatic inflation of the variance makes sense. The contemporaneous spatial dependence existing in the regional income systems invalidates the $i . i . d$ assumption. The effective sample size for the transition probability estimator $\hat{P}$ is less than $N \times T$ . Thus, the actual variance of each element of $\hat{P}$ should be larger than what is indicated in equation (6). Since all of the three mobility estimators are derived from P , their variances would also be inflated.

Properties of Two-sample Test Statistics

Results regarding the properties of the two-sample test statistics for the three mobility measures are discussed in this section.

Size

The general pattern for the impact of the contemporaneous spatial dependence on the size properties for three mobility measures is quite similar. Thus, we focus only on discussing the results for Shorrocks mobility measure.

The rejection ratios of the null under various circumstances for the two-sample test statistics for the Shorrocks mobility measure are displayed in Figure 2. The x-axis of each subplot is indexed by $ρ^{(A)}$ , the level of contemporaneous spatial dependence in system A, and the y-axis indexes the rejection ratio of the null. The upper and lower bounds of the 95 percent confidence interval $[0.0365, 0.0635]$ are shown by two black horizontal dashed lines. The upper-, lower-, and the two-tail test are symbolized in blue, green, and red lines, respectively. $ρ^{(B)}$ , the contemporaneous spatial dependence in system B, becomes stronger from the left to the right subplot. From the top subplot to the bottom, the sample size increases. We can easily observe that relatively strong spatial dependence in either distribution (such as $ρ^{(A)} = 0.7$ or $ρ^{(B)} = 0.7$ ) has a significant influence on the size properties. It tends to make the size biased upward. As the level of spatial dependence in either system becomes higher, the upward bias tendency becomes stronger. It also seem to be the case that larger sample size is companied with more upward biased size. Comparing three different alternatives, the upper- and lower-tests seem to be more robust to spatial dependence than the two-tail test. This is especially true when $ρ^{(A)}$ or $ρ^{(B)}$ is quite large.

Figure 2.

Size properties of the upper-, lower-, and two-tail two-sample tests for $M_{1} (P)$ .

Figure A3 in Appendix A shows the impact of contemporaneous spatial dependence of varied levels on the size properties of the two-sample test statistics for the determinant mobility measure, while Figure A4 for the eigenvalue mobility measure. The patterns are rather similar to what we have observed for the Shorrocks mobility measure.

Power

Turning to the power of the test statistics, we see similarity across the three mobility measures. To save the space, we are only going to discuss results for the Shorrocks mobility measure in detail.⁴

Figure 3 displays the rejection ratios when the mobility difference between two income systems is small ( $β = 0.01$ ). Since the true mobility difference is negative, rejection ratios of the lower- and two-tail tests shed light on their power properties, while the ratios of the upper-tail test indicate its robustness as it is not supposed to pick up the negative difference. The power for the lower- and two-tail tests tends to grow with the sample size: for the lower-tail test, the rejection ratio increases from $0.146$ all the way to $0.957$ when both systems do not suffer from spatial dependence. The reason is that the variance for each of the mobility in the z-test statistic decreases with the sample size $N, T$ . Therefore, the denominator, which is the difference between the standard deviations for mobilities measured for two economic systems, decreases with the sample size. Thus, facing the same mobility difference, the test with a larger set of observations tends to reject more. The general pattern for the impacts of spatial dependence also varies between small and large sample size. Looking at the first row where sample size is fairly small $N = 25, T = 50$ , it seems that the power for the two-tail test increases with the spatial dependence level in either system, while the power for the lower-tail test increases with the spatial dependence level in income system B and decreases with the spatial dependence level in system A. This is also true for some larger sample cases $N = 25, T = 200$ and $N = 169, T = 50$ . However, when sample size is quite large as shown in the bottom row, the power decreases with stronger spatial dependence in either system. For the upper-tail test, the rejection ratios are always close to 0 except when spatial dependence is strong in either system and sample size is relatively small.

Figure 3.

Power properties of the upper-, lower-, and two-tail two-sample tests for $M_{1}$ ( $β = 0.01$ ).

Increasing the difference between two transition probability matrices ( $β = 0.03$ ) results in a stronger mobility difference of $- 0.068$ for Shorrocks mobility measure. As shown in Figure 4, the power for both of the lower- and two-tail tests mildly increases with the spatial dependence level in income system B and decreases with the spatial dependence level in system A when sample size is very small, $N = 25, T = 50$ . For larger sample size, both tests have good power properties. They become less powerful in detecting the mobility difference when the spatial dependence is stronger in either system. However, as the sample size becomes larger, the decreasing trend is more and more negligible. Looking at the third row, it is clear that the power does not decrease until the dependence is very strong ( $ρ^{(A)} = 0.9$ or $ρ^{(B)} = 0.9$ ).

Figure 4.

Power properties of the upper-, lower-, and two-tail two-sample tests for $M_{1}$ ( $β = 0.03$ ).

Turning to the power properties of the tests when the mobility difference is much larger ( $- 0.110$ ), the patterns are more consistent as shown in Figure 5. Only when the sample size is quite small, does the power decreases as the spatial dependence level in either system increases. This decreasing trend can be readily ignored when sample size is large: the power is quite close to 1 even when spatial dependence is strong. The impact of the spatial dependence is very similar for the other two mobility measures.

Figure 5.

Power properties of the upper-, lower-, and two-tail two-sample tests for $M_{1}$ ( $β = 0.05$ ).

Adjusting Critical Values

As shown in the last section, contemporaneous spatial dependence inflates variances of sampling distributions of mobility estimators and raises the Type I error rates for both one-sample and two-sample tests. We resort to adjusting critical values to their “true” levels in order to maintain a proper size for the tests. Since we adopted Monte Carlo simulations to simulate the null where (1) mobility level equals a given level for the one-sample test and (2) two regional system are equally mobile for the two-sample test, the empirical sampling distribution of estimates could be considered as the “true” sampling distribution to the presence of spatial autocorrelation of varying levels. Thus, the “true” critical values at the 5 percent significance level for a two-sided test are the 25th and 975th of the ordered 1,000 estimated test statistics.

One-sample Test

For the one-sample test in equation (12), assigning the “true” mobility level which is used as a simulation parameter (as shown in second column [ $M^{(A)}$ ]of Table 1) to $χ$ would give $z_{1}$ estimates under the null. Therefore, the $z_{1}$ statistics estimated from $1, 000$ realizations should follow the standard normal distribution $N (0, 1)$ . By testing those estimates against $N (0, 1)$ , we could know whether the empirical distribution deviates significantly from $N (0, 1)$ and thus whether adjustments are needed.

Focusing on the Shorrock mobility measure, we plot the upper and lower empirical critical values for its one-sample test where testing for $N (0, 1)$ is rejected in Figure 6. Similar to before, each subplot represents a specific sample size and the x-axis indexes contemporaneous spatial autocorrelation level ( $ρ$ ). From the plot, we could discern that adjustment is needed for all cases when sample size is small. On the opposite, for a large sample size as shown in the lower-right subplot, the critical values $- 1.96$ and $1.96$ obtained from $N (0, 1)$ could well serve the purpose for regional systems that are not highly spatially autocorrelated ( $ρ < 0.5$ ). However, strong spatial autocorrelation inflates critical values more severely for larger sample sizes. Results for the other two mobility measures are similar and are available upon request.

Figure 6.

Empirical critical values of a one-sample two-tail test for $M_{1} (P)$ .

Two-sample Test

Turning to the two-sample test (equation [13]), since the test statistic $z_{2}$ follows a standard normal distribution asymptotically, we adopt a similar approach. That is, we test for the standard normal distribution and obtain empirical critical values for cases where the tests are rejected. Those empirical critical values are visualized in Figure 7. The plots suggest that when both regional systems are strongly spatially autocorrelated, the critical values have to be increased for the comparison to be statistically valid. What’s more, the inflation of critical values gets more severe with the increasing spatial autocorrelation level in either system. If both regional systems are weakly spatially autocorrelated, there is no need to make adjustment.⁵

Figure 7.

Empirical critical values of a two-sample two-tail test for $M_{1} (P)$ .

Conclusion

Regional income mobility measures are useful complements to regional inequality measures as combined they allow for a fuller understanding of regional income systems and their dynamics. However, potential interactions between regions invalidate the $i . i . d$ assumption underlying tests of mobility in Markovian frameworks. This challenge is rather pertinent in the regional context as the notion of spatial dependence being a rule instead of an exception is widely acknowledged. This article takes up the challenge and explores the impacts of spatial dependence on the mobility inference via a series of Monte Carlo experiments.

We focused on three Markov-based mobility measures and found that the impacts from spatial dependence are rather similar. Dependence does have a major influence on the properties of the mobility estimators, one-sample, and two-sample test statistics. It does not bias the mobility estimators when the spatial dependence is not extreme but does dramatically increase the variances, leading to a inflated Type I error rate for a one-sample test. As for the two-sample test, the size tends to become more and more upward biased with increasing spatial dependence in either income system, which indicates that conclusions about differences in mobility between two different regional systems need to drawn with caution as the presence of spatial dependence can lead to false positives. The reason for the size distortion is due to the inflated variance of the test statistics.

For the power properties, the impact has a mixed pattern in small sample settings, while when sample size is large, the power decreases with stronger spatial dependence. Since the size is upward biased when there is spatial dependence in either income system, the power acquired based on the theoretical critical value would be inflated. Therefore, the actual power under the impact of spatial dependence is quite low.

Having found that spatial dependence impacts on the properties of mobility estimators and related tests, we attempted to account for the dependence by making adjustments to the critical values based on the results acquired from the Monte Carlo experiments. We have also tested the empirical distributions of the test statistics against their analytical asymptotic distribution to differentiate cases where the impact of spatial autocorrelation is so trivial that an adjustment is not needed. It turns out that there is no need to make adjustment under the circumstance of a relatively large sample size and weak spatial dependence. Further research could be directed to the generalization of the adjustments to incorporate a wider range of cases. Empirical applications of the adjusted one-sample and two-sample tests are of great potential once a general formula is readily available.

Footnotes

Appendix A

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: The authors received financial support from National Science Foundation Grant SES-1421935.

ORCID iD

Wei Kang

Notes

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