Time-Series Based Empirical Assessment of Random Urban Growth: New Evidence from France

Abstract

Modern urban growth literature frequently uses unit-root tests in order to check the empirical relevance of Gibrat’s law of random growth. The contradictory nature of the test results provided by this literature is most likely linked to the low power of unit-root tests. To address this problem, we apply unit-root testing to a large-sized sample of high-quality French census data covering an exceptionally long time span of more than two centuries. We add subsequent cointegration tests in order to detect the possible presence of cointegrated random growth, which may reflect the fact that cities with a similar economic structure react fairly similarly to exogenous growth shocks. According to the test results, the random growth hypothesis cannot be rejected for a very large majority of the tested French cities; on the other hand, the null hypothesis of absence of cointegration cannot be rejected in more than 95% of the cases. Our findings therefore provide empirical support for non-cointegrated random growth.

Appendix: Cointegration of \({ln} S_{{i,t}}\) and \({ln} S_{{j,t}}\) Implies Growth Process (5)

Assume cointegration of $ln S_{i,t}$ and $ln S_{j,t}$, where growth factor shocks $\gamma _{i,t}$ and $\gamma _{j,t}$ are iid distributed over time (but not necessarily across cities), with a density distribution $f (\gamma )$ characterized by $E[\gamma ] = \mu _{\gamma }$ and $Var[\gamma ] = \sigma _{\gamma }^{2}$ verifying $\vert \mu _{\gamma } \vert < \infty$ and $0< \sigma _{\gamma }^{2} < \infty$. As cointegrated variables must be I(1) , the functional form of $S_{i,t}$-growth writes:

$$\begin{aligned} S_{i,t} =S_{i,t-1} \; \gamma _{i,t}. \end{aligned}$$

(A.1)

In empirical applications, we have necessarily an initial observation $S_{i,0}$, so we can rewrite equation (A.1) as follows:

$$\begin{aligned} S_{i,t} = \gamma _{i,t} \times \gamma _{i,t-1} \times \ldots \times \gamma _{i,1} \times S_{i,0}. \end{aligned}$$

(A.2)

In the same way, we obtain for city j:

$$\begin{aligned} S_{j,t} = \gamma _{j,t} \times \gamma _{j,t-1} \times \ldots \times \gamma _{j,1} \times S_{j,0}. \end{aligned}$$

(A.3)

Cointegration further implies that there is a linear combination of $ln S_{i,t}$ and $ln S_{j,t}$ which is I(0) , and as such characterized by time-independence of the first two theoretical moments. So we have to find a way to link (A.2) and (A.3) in a manner that enables us to formulate a linear combination of $ln S_{i,t}$ and $ln S_{j,t}$. A general way of doing that is to raise (A.2) and (A.3) to powers $\delta _{1}$ and $\delta _{2}$, with $[\delta _{1} \; \delta _{2}]' \ne [0 \; 0]'$, to divide the powered (A.2) by the powered (A.3), and then to take natural logarithms. We get the cointegration equation

$$\begin{aligned} x_{ij,t} = \delta _{1} ln S_{i,t} - \delta _{2} ln S_{j,t} - \alpha _{ij,0} \end{aligned}$$

(A.4)

where $\alpha _{ij,0} = \delta _{1} ln S_{i,0} - \delta _{2} ln S_{j,0}$ has a natural interpretation as the difference between the logs of initial population levels of cities i and j, and with $x_{ij,t} = (\delta _{1} ln \gamma _{i,t} - \delta _{2} ln \gamma _{j,t}) + (\delta _{1} ln \gamma _{i,t-1} - \delta _{2} ln \gamma _{j,t-1}) + \cdots + (\delta _{1} ln \gamma _{i,1} - \delta _{2} ln \gamma _{j,1})$.

Due to time-independence of the first theoretical moments, we have

$$\begin{aligned} E[x_{ij,t}] = \mu _{x}, \quad \forall t \end{aligned}$$

(A.5)

and

$$\begin{aligned} Var[x_{ij,t}] = \sigma _{x}^{2},\quad \forall t. \end{aligned}$$

(A.6)

Now define the process $\lbrace \omega _{ij,t} \rbrace$:

$$\begin{aligned} \omega _{ij,t} = \delta _{1} ln \gamma _{i,t} - \delta _{2} ln \gamma _{j,t}. \end{aligned}$$

(A.7)

$\omega _{ij,t}$ is a linear combination of (log transformed) iid processes, so it is itself iid, with mean $E[\omega _{ij,t} ] = \mu _{\omega }$ and variance $Var[\omega _{ij,t}] = \sigma _{\omega }^{2}$ verifying $\vert \mu _{\omega } \vert < \infty$ and $0< \sigma _{\omega }^{2} < \infty$. We can now rewrite $x_{ij,t}$ as follows:

$$\begin{aligned} x_{ij,t} = \omega _{ij,t} + \omega _{ij,t-1} + \ldots + \omega _{ij,1} \end{aligned}$$

(A.8)

implying

$$\begin{aligned} E[x_{ij,t}] = t \times \mu _{\omega } \end{aligned}$$

(A.9)

and

$$\begin{aligned} Var[x_{ij,t}] = t \times \sigma _{\omega }^{2}. \end{aligned}$$

(A.10)

So we have $\mu _{x} = t \times \mu _{\omega }$ and $\sigma _{x}^{2} = t \times \sigma _{\omega }^{2}$ implying $\mu _{x} = \mu _{\omega } = 0$ and $\sigma _{x}^{2} = \sigma _{\omega }^{2} = 0 \; \forall t > 0$. Moreover, we necessarily have $\delta _{1} = \delta _{2}$, because $E[ln \gamma _{i,t}] = E[ln \gamma _{j,t}]$ and

$$\begin{aligned} E[\omega _{ij,t}] = \mu _{\omega } = \delta _{1} E[ln \gamma _{i,t}] - \delta _{2} E[ln \gamma _{j,t}] = 0. \end{aligned}$$

(A.11)

Finally, with $E[\omega _{ij,t}] = Var[\omega _{ij,t}] = 0$, we have $\omega _{ij,t} = 0, \; \forall t$, implying

$$\begin{aligned} \delta _{1} ln \gamma _{i,t} = \delta _{2} ln \gamma _{j,t} \Leftrightarrow \gamma _{i,t} = \gamma _{j,t}, \quad \forall t. \end{aligned}$$

(A.12)

Growth shocks affecting cities i and j at a given time are thus identical ($\gamma _{i,t} = \gamma _{j,t} = {\bar{\gamma }}_{t}, \forall t$), and sizes of i and j are determined by (5).