Exploring the dynamics of business survey data using Markov models

Abstract

Business tendency surveys are widely used for monitoring economic activity. They provide timely feedback on the current business conditions and outlook. We identify the unobserved macroeconomic factors behind the distribution of quarterly responses by Austrian firms on the questions concerning the current business climate and production. The aggregate models identify two macroeconomic regimes: upturn and downturn. Their dynamics is modeled using a regime-switching matrix. The micro-founded models envision dependent responses by the firms, so that a favorable or an adverse unobserved common macroeconomic factor increases the frequency of optimistic or pessimistic responses. The corresponding conditional transition probabilities are estimated using a coupling scheme. Extensions address the sector dimension and introduce dynamic common tendencies modeled with a hidden Markov chain.

Appendix: Likelihood functions and constraints

All estimators for the above models are obtained by maximizing the logarithm of a likelihood function subject to constraints. Since static models are particular cases of dynamic models, we will focus on the latter while indicating how a static model is obtained from a dynamic one.

We begin with the aggregate model based on a regime-switching matrix $\mathcal{P}$. Let m be the number of possible responses to a survey question (we have $m=3$). The likelihood function is

$$\begin{aligned}&L(P^U,P^D,\mathcal{P}, p)\\&\quad =\prod _{t=1}^{T}\left\{ [(p,\,1-p)\mathcal{P}^{t-1}]_1\prod _{i=1}^m\prod _{j=1}^m (P^U_{i,j})^{n^t(i,j)} \right. \left. + [(p,1-p)\mathcal{P}^{t-1}]_2 \prod _{i=1}^m\prod _{j=1}^m (P^D_{i,j})^{n^t(i,j)}\right\} . \end{aligned}$$

The term $[(p,\,1-p)\mathcal{P}^{t-1}]_k$ is the k-th coordinate of the vector $(p,\,1-p)\mathcal{P}^{t-1}$, where

$$\begin{aligned} \mathcal{P}=\left( \begin{array}{cc} \alpha &{}\quad 1-\alpha \\ 1-\beta &{}\quad \beta \\ \end{array} \right) . \end{aligned}$$

The linear equality constraints are

$$\begin{aligned} \sum _{j=1}^m P^U_{i,j}=1, \quad \sum _{j=1}^m P^D_{i,j}=1, \quad i=1,2,\ldots ,m, \end{aligned}$$

whereas the linear inequality constrains are given by

$$\begin{aligned}&P^U_{i,j}\ge P^D_{i,j}\;\text { if} \;j<i \;\text { and}\; P^D_{i,j}\ge P^U_{i,j}\;\text { if} \;j>i,\quad i=1,2,\ldots ,m; \\&\sum _{j=1}^{i-1}(P^U_{i,j}-P^D_{i,j})+\sum _{j=i+1}^{m}(P^D_{i,j}-P^U_{i,j})\ge \epsilon _i,\quad i=1,2,\ldots ,m. \end{aligned}$$

The thresholds $\epsilon _i$ are given non-negative numbers. If $\epsilon _i$ vanishes for some i, then the corresponding inequality in the second group follows from the inequalities involving i from the first group. The values $P^U_{i,j}$, $P^D_{i,j}$, p, $\alpha$ and $\beta$ must belong to [0, 1]. The dynamic setting reduces to the static one when $\mathcal{P}$ becomes a $2\times 2$ identity matrix $I_2$.

Turning to the micro-founded model, let us first generalize the formulas for the conditional probabilities to any number of states m. Let P be $m\times m$ transition matrix with entries $P_{i,j}$. Assuming that $1\succ 2\succ \cdots \succ m$, set

$$\begin{aligned} P_{i,j}(1) =\left\{ \begin{array}{cl} \frac{1}{P_{i}}P_{i,j} &{}\quad \text { if }j<i, \\ \frac{\Delta _i}{P_{i}}P_{i,i} &{}\quad \text { if }j=i, \\ 0 &{}\quad \text { if }j>i; \end{array} \right. \text { and } P_{i,j}(0)=\left\{ \begin{array}{cl} \frac{1}{1-P_{i}}P_{i,j} &{}\quad \text { if }j>i, \\ \frac{1-\Delta _i}{1-P_{i}}P_{i,i} &{}\quad \text { if }j=i, \\ 0 &{}\quad \text { if }j<i. \end{array} \right. \end{aligned}$$

Here, $P_i=P_{i,1}+P_{i,2}+\cdots +P_{i,i-1}+\Delta _iP_{i,i}$, $0\le \Delta _i\le 1$, $i=1,2,\ldots ,m$. Depending on whether $\Delta _i$ is larger or smaller than $\Delta _i^*=\frac{P_{i,1}+P_{i,2}+\ldots +P_{i,i-1}}{1-P_{i,i}}$, $P_{i,i}(1)\;(P_{i,i}(0))$ will be larger (smaller) or smaller (larger) than $P_{i,i}$, $2\le i\le m-1$. The probabilities of moving to the better (worse) states, $j<i$ ($j>i$) increase relative to $P_{i,j}$ under favorable (adverse) macroeconomic conditions given by $\chi _i=1$ ($\chi _i=0$) for the firms belonging to class i. Each transition probability is multiplied by a factor that exceeds one. The adjustment of these probabilities according to macroeconomic conditions depends on the parameters $\Delta _i$, where $P_{i,i}$ is the probability that a firm will not change its opinion in the next quarter. There is no margin for the adjustment for $P_{1,1}$ and $P_{m,m}$. $\Delta _1=1$ and $\Delta _m=0$ because $P_{1,i}(1)$ must be 0 for all $i>1$ and $P_{m,j}(0)$ must be 0 for all $j<m$. The remaining $\Delta _i$ are estimated, together with the remaining model parameters, as a vector $\vec {d}$ with $m-2$ coordinates, such that $\Delta _i=d_{i-1}$. The percentage of variation of $P_{j,j}(\chi _j)$, $2\le j\le m-1$, can be expressed as $\left( \frac{d_{j+1}}{P_j}-1\right) \cdot 100$ if $\chi _j=1$ and $\left( \frac{1-d_{j+1}}{1-P_j}-1\right) \cdot 100$ if $\chi _j=0$.

The likelihood function of the micro-founded setting is given by:

$$\begin{aligned} L(\vec {\rho },q,\vec {d},\mathcal{P})=\prod _{t=1}^{T} \sum _{l=1}^{n_\mathbf{{BV}}}[\vec {\rho }^{(l)}\mathcal{P}^{t-1}]_i \prod _{s=1}^{S} \prod _{i=1}^m \prod _{j=1}^m [q_{i,s}P_{i,j}+ (1-q_{i,s})P_{i,j}(\chi _i^{(l)})]^{n^{(s)}_{i,j}(t)}. \end{aligned}$$

In the above formula, $n_\mathbf{{BV}}$ denotes the number of elements in a set $\mathbf {BV}$ of binary vectors with m coordinates, and S stands for the number of industry sectors considered. We set $S=1$ in the case of no sector differentiation, so that q becomes a vector. For the static model, $\mathbf {BV}$ coincides with the set $\mathbf {\{0,1\}^m}$ of all binary vectors with m coordinates. If $\mathbf {BV}= \mathbf {\{0,1\}^m}$, then the numbering convention of Sect. 4 applies, i.e. the vector $(1, 1, \ldots , 1)$ is numbered by 1, while $(0, 0, \ldots , 0)$ receives the number $2^m$. Otherwise, if $\mathbf {BV}\subset \mathbf {\{0,1\}^m}$, then the binary vectors $\vec {\chi }\in \mathbf {\{0,1\}^m}$ must be numbered according to $\vec {\chi }^{(l)}$, $l=1,2,\ldots ,n_\mathbf{{BV}}$. For example, the estimates reported in Sect. 6 are obtained assuming that $\mathbf {BV}$ contains four vectors that are numbered in descending order of the probabilities assigned to them by the solution of the respective static model. The l-th coordinate $\rho _l$ of the $n_\mathbf{{BV}}$-vector $\vec {\rho }$ equals the probability assigned to the binary vector $\vec {\chi }^{(l)}$. The entry $\mathcal{P}_{l,k}$ of the $n_\mathbf{{BV}}\times n_\mathbf{{BV}}$ matrix $\mathcal{P}$ is the probability that the macroeconomic scenario encoded by $\vec {\chi }^{(l)}$ will be followed by the macroeconomic scenario corresponding to $\vec {\chi }^{(k)}$.

Linear inequality constraints read $\mathcal{P}_{l,k}$, $q_{i,s}$, $\rho _l$, $d_j\in [0,1]$. Linear equality constraints are

$$\begin{aligned} \sum _{l=1}^{n_\mathbf{{BV}}}\rho _l=1, \quad \sum _{k=1}^{n_\mathbf{{BV}}}\mathcal{P}_{l,k}=1,\quad i=1,2,\ldots ,{n_\mathbf{{BV}}}. \end{aligned}$$

The nonlinear (with respect to $\mathcal{P}$) equality constraints are given by

$$\begin{aligned} \sum _{l=1}^{n_\mathbf{{BV}}}[\vec {\rho }^{(l)}\mathcal{P}^{(t-1)}]_i\chi ^{(l)}_i=P_i,\quad i=1,2,\ldots ,m,\quad t=2,3,\ldots ,T. \end{aligned}$$

Recall that $P_i$ contains the parameter $d_{i-1}$, $i=2,3,\ldots ,m-1$. The dynamic setting reduces to the static one when $\mathbf {BV}=\mathbf {\{0,1\}^m}$ and $\mathcal{P}$ equals a $2^m\times 2^m$ identity matrix $I_{2^m}$. A dynamic setting involving a subset $\mathbf {BV}$ of all possible macroeconomic scenarios reduces to the dynamic setting with the whole set $\mathbf {\{0,\,1\}^m}$ if, a) non-zero coordinates of $\vec {\rho }$ correspond to scenarios belonging to $\mathbf {BV}$, b) the diagonal entries of $\mathcal{P}$ are equal to 1 for the states from $\mathbf {\{0,\,1\}^m}{\setminus } \mathbf {BV}$, and c) the remaining non-zero entries of $\mathcal{P}$ correspond to couples of states from $\mathbf {BV}$.