Dissecting Brazilian agriculture business cycles in high-dimensional and time-irregular span contexts

Abstract

Business cycle analysis faces the challenge of high-dimensional databases with a time-irregular span. This study addresses these issues to estimate the Brazilian agricultural business cycle by proposing the use of an entropic test of relative information and the estimation of a generalized dynamic factor model in the context of a time-irregular span. In addition, we assess the co-movements between the estimated cycle and a wide range of weather, macroeconomic, and sectoral variables. The main results are: (i) the sharpest crises of the agricultural cycle are associated more with weather events than with Brazil’s economic crises, reinforcing that agriculture has a stabilizing role in the aggregated cycles; (ii) among all the variables analyzed, those related to weather (minimum and maximum temperature and precipitation) present the greatest commonalities with the agricultural cycle and are leading indicators; (iii) minimum temperatures and precipitation in states are essentially pro-cyclical, while maximum temperatures are pro-cyclical in typically colder states and counter-cyclical in warmer states; (iv) following the weather variables, credit variables had the highest average commonalities, with phases of expansion in the agricultural cycle leading to contractions in the indicators of arrears and defaults and in the interest rate of rural credit, with some time lag; (v) the behavior of the Brazilian economy and global demand for imports are also antecedent and pro-cyclical, but with relatively smaller commonalities; and (vi) expansions of the agricultural cycle lead to contractions in the unemployment rate of the economy. The contribution of this study is twofold: (1) its methodological innovation and (2) its application to the Brazilian agricultural business cycle, both of which are unprecedented.

Appendix

1.1 Entropy relative information

First, as den Reijer (2010) presented, we can define $f_{1}(\tilde{x}):\tilde{x}\sim \textit{N}_{N}(0,\Gamma ), \text {with } \Gamma = C \Lambda C^{'}$ as the density function of an N-dimensional data vector x⁶ such that $f_{1}(x):x \sim \textit{N}_{N}(0,\Lambda )$ and $f_{2}(\tilde{x}):\tilde{x}\sim \textit{N}_{N}(0,I)$, such that $f_{2}(x): x \sim \textit{N}_{N}(0,I)$, where $x = C^{'}\tilde{x}$. The Kullback-Leibler numbers are defined as (3):

$$\begin{aligned} G_{1} = E_{f_{1}}\left( \log \left( \frac{f_{1}(x)}{f_{2}(x)} \right) \right) \text {and} G_{2} = E_{f_{2}}\left( \log \left( \frac{f_{2}(x)}{f_{1}(x)}\right) \right) \end{aligned}$$

(3)

where $G=G_{1}+G_{2}$ is the measure of information used to discriminate between the two density functions, with $G=0$ when $f_{1}(x)=f_{2}(x)$ and $G=\infty$ when we have perfect discrimination (Golan and Maasoumi 2008; Young and Calvert 1974; Burnham and Anderson 2002). Note that $tr(\Lambda )=tr(\Gamma )=N$. We have $G_{1}=-\log \det (\Lambda )$ and $G_{2}=\log \det (\Lambda )+ \frac{1}{2}(tr(\Lambda ^{-1})-N)$, and thus, $2G_{2} = 2\log \det (\Lambda ) + (tr(\Lambda ^{-1})-N), 2G_{1} = -2\log \det (\Lambda )$ and then, (4):

$$\begin{aligned} 2G= & {} tr(\Lambda ^{-1})-N = tr(\Lambda ^{-1})+tr(\Lambda )= \sum _{j=1}^{N} \frac{(1-\lambda _{j}^{2})}{\lambda _{j}} \end{aligned}$$

(4)

Therefore, G is small (not discriminating) if the eigenvalues $\lambda _{j}$ are close to 1 but become large (discriminating) for “small” eigenvalues. In a Gaussian case, alternative measures of entropy and information can be used. For this purpose, we define $x_{t}$ as an N-dimensional vector of observed data at time $t=1,...,T$ normalized and normally distributed with zero mean and variance, so that $x\sim N(0,\Gamma )$ with $E(x_{t}x_{t}^{'})=\Gamma$, $tr(\Gamma )=N$. Entropy as a measure of disorder for a stationary, normally distributed vector can be defined as in Golan and Maasoumi (2008): $2\mathbb {E}_{x}=cN+\log \det (\Gamma )$, with $c=\log (2\pi )+1\approx 2.84$, where $2\mathbb {E}_{x,max}=cN$ when $\Gamma =I_{N}$ (Golan and Maasoumi 2008; Goodwin and Payne 1977). Therefore, the information or negentropy is defined as (5):

$$\begin{aligned} 2Inf_{x}=2(\mathbb {E}_{x,max}-\mathbb {E}_{x})=-\log \det (\Gamma ) \ge 0 \end{aligned}$$

(5)

If $\Gamma =I_{N}$, we have a zero value. Considering all the details, we can then define the relative information (RI) as in (6).

$$\begin{aligned} RI_{(N,t)}= & {} \frac{2\mathbb {E}_{x,max}-\mathbb {E}_{x(N)}}{2\mathbb {E}_{x,max}} \nonumber \\= & {} \frac{2Inf_{x}}{2\mathbb {E}_{x,max}} \nonumber \\= & {} \frac{2Inf_{x}}{cN} \end{aligned}$$

(6)

Note that if $\mathbb {E}_{x(N)}=\mathbb {E}_{x,max}$, then $RI_{N}=0$ and $\mathbb {E}_{x(N)}=0 \Rightarrow RI_{N}=1$.

1.2 Entropic relative information measure in factor model context

This subsection explores the link between $RI_{N}$ and factor models (static or dynamic). First, we should consider $x_{it}=(x_{1t},...,x_{nt})^{'}$ with $n \in N,t \in T$ as a stationary N-dimensional vector process with zero mean driven by k factors, as in (7).

$$\begin{aligned} x_{it}= & {} \chi _{it}+\xi _{it} = \sum _{j=1}^{N}b_{ij}(L)u_{jt}+\xi _{it}\nonumber \\= & {} B_{N}F_{t}+\epsilon _{t}, x_{it}\in \mathbb {R}^{N}, F_{t}\sim \textit{N}_{k}(0,I_{k}), \epsilon _{t}\sim \textit{N}_{k}(0,\Psi _{11}) \end{aligned}$$

(7)

where $\chi _{it}$ is the common component, $\xi _{it}$ is the idiosyncratic component, $b-{ij}(L)=B_{N}=B_{0}^{n}+B_{1}^{n}L+...+B_{s}^{n}L^{s}$ represents the (dynamic) loadings of order s, $u_{ij},$ with $j=1,...,q;t=1,...,T$ are common shocks of mutually orthogonal white noise processes with unit variance. The variance between the first N elements of $x_{it}$ is equal to $\Gamma (N)=B_{N}B_{N}^{'}+\Psi _{11}$ as demonstrated in Bai and Ng (2008).

When we add variable $x_{N+1,t}$, we obtain (8).

$$\begin{aligned} {x_{it}\atopwithdelims ()x_{N+1,t}}={B_{N}\atopwithdelims ()b_{N+1}}F_{t}+{\epsilon _{t} \atopwithdelims ()\epsilon _{N+1,t}} \end{aligned}$$

(8)

with covariance $\Gamma (N+1) =\left( \begin{array}{cc}\Gamma (N) &{} \Gamma _{12} \\ \Gamma _{21}&{} \Gamma _{1} \\ \end{array} \right)$, where $\Gamma _{12}=B_{N}b_{N+1}^{'}+\Psi _{12}$ and $\Psi _{12}=E(\epsilon _{t}\epsilon _{N+1,t})$.

Because $x_{it}$ is normalized, we have $b_{N+1}b_{N+1}^{'}+\sigma _{N+1}^{2}=1$, with $\sigma _{N+1}^{2}=E(\epsilon _{N+1,t}^{2})$. Using the rule of determinants for partitioned matrices, we obtain $\det (\Gamma (N+1))= \det (\Gamma (N))(1-a_{N+1})$, where $a_{N+1}=b_{N+1}B_{N}^{'}+\Psi _{12})\Gamma (N)^{-1}(B_{N}b_{N+1}^{'}+\Psi _{12})$ and $0\le (1-a_{N+1})\le 1$. Considering these results:

$$\begin{aligned}{} & {} RI_{(N,t)}= \frac{-logdet(\Gamma (N))}{cN}\\{} & {} RI_{(N+1,t)}= \frac{-logdet(\Gamma (N+1))}{c(N+1)} = \frac{-logdet(\Gamma (N)+(1-a_{N+1}))}{c(N+1)}, \end{aligned}$$

The relationship between $RI_{(N+1,t)}$ and $RI_{(N,t)}$ is expressed in (9):

$$\begin{aligned} RI_{(N+1,t)}=RI_{(N,t)}- \frac{1}{N+1}\left( \frac{\log (1-a_{N+1})}{c}+RI_{(N,t)}\right) \end{aligned}$$

(9)

Therefore, relative information is added only if $RI_{N+1,t}>RI_{(N,t)}$ or if $-log(1-a_{N+1})>cRI_{(N,t)} \Rightarrow E(x_{N+1,t} x_{N,t}^{'})=(b_{N+1}B_{N}^{'}+\Psi _{12} \ne 0)$. The last condition was used to apply a formal statistical test. From Eq. 9, we have that $RI_{(N+1,t)}=RI_{(N,t)}$ if $a_{N+1}=1-\exp {(-cRI_{(N,t)})}$. Whenever $RI_{(N,t)}$ is close to zero, $RI_{(N+1,t)}$ increases for relatively small values of $a_{N+1}$. When it is close to one, $a_{N+1}$ should be close to one to allow $x_{N+1,t}$ to add relative information.

We can simplify Eq. 9 considering $\Gamma (N)=C\Lambda C^{'}$ and the linear transformation $\tilde{x}_{t}=U^{'}\Lambda ^{-\frac{1}{2}}C^{'}x_{t}$ and $\tilde{x}_{N+1,t}=\upsilon ^{-1}x_{N+1,t}$, where U orthogonal and $\upsilon ^{2}=1$ obtained by singular value decomposition with $\Lambda ^{-\frac{1}{2}}C^{'}\Gamma =U\Sigma \upsilon$ where $\Sigma =(\phi ,0,...,0)^{'}$, from which we also have $\Gamma _{12}=0 \Rightarrow \Sigma =0$. Therefore, we have (10)

$$\begin{aligned} {\tilde{x}_{N,t} \atopwithdelims ()\tilde{x}_{(N+1,t)}} = \textit{N}_{N} \left( {0 \atopwithdelims ()0}, \tilde{\Gamma }(N+1)\right) \end{aligned}$$

(10)

where ${\Gamma }(N+1) =\left( \begin{array}{cc} I(N) &{} \Sigma \\ \Sigma &{} 1 \\ \end{array}\right)$, $det(\tilde{\Gamma }(N+1))=det(I_{N})(1-\phi _{1}^{2})\Rightarrow RI_{(N+1,t)}= \frac{-\log (1-\phi _{1}^{2})}{c(N+1)}$, and $\phi _{1} \in [0,1]$ is the coefficient of the canonical correlation. Because $RI_{(N,t)}=0$ from the hypothesis in Eq. 9 so that there is a gain in relative information, we have $\tilde{RI}_{(N+1,t)}= \frac{-log(1-\phi _{1}^{2})}{c(N+1)}$.

Table 5 Chronology of Brazilian extreme climate events

Table 6 Description—database sources