Let $\left{q_{i t}\right}$ be a balanced panel ${ }^{22}$ of $N$ time-series with $T$ observations which are generated by $$ \Delta q_{i t}=\delta_{i} t+\beta_{i} q_{i t-1}+u_{i t} $$ where $-2<\beta_{i} \leq 0$, and $u_{i t}$ has the error-components representation $$ u_{i t}=\alpha_{i}+\theta_{t}+\epsilon_{i t} $$ $\alpha_{i}$ is an individual-specific effect, $\theta_{t}$ is a single factor common time effect, and $\epsilon_{i t}$ is a stationary but possibly serially correlated idiosyncratic effect that is independent across individuals. For each individual $i, \epsilon_{i t}$ has the Wold moving-average representation $$ \epsilon_{i t}=\sum_{j=0}^{\infty} \theta_{i j} \epsilon_{i t-j}+u_{i t} $$
The Im, Pesaran and Shin test as well as the Maddala-Wu test (discussed below) relax the homogeneity restrictions under the alternative hypothesis. Here, the null hypothesis $$ H_{0}: \beta_{1}=\cdots=\beta_{N}=\beta=0, $$ is tested against the alternative $$ H_{A}: \beta_{1}<0 \cup \beta_{2}<0 \cdots \cup \beta \beta_{N}<0 . $$ The alternative hypothesis is not $H_{0}$, which is less restrictive than the Levin-Lin alternative hypothesis.
$$ \pi_{u t}=u_{u t} Y_{u t}-g r_{i t} K_{u}=\left(u_{i t} \phi-g r_{u t}\right) K_{u t} $$ and expected profit is $E\left(\pi_{i t}\right)=\left(\phi-g r_{i t}\right) K_{i t}$. In this economy, firms may go bankrupt as soon as their net worth becomes negative, that is $A_{i t}<0$. The law of motion of $A_{i t}$ is: $$ A_{u}=A_{u t-1}+\pi_{u \prime} $$ that is, net worth in previous period plus (minus) profits (losses). Making use of (4.16) and (4.17), it follows that the bankruptcy state occurs whenever: $$ u_{i t}<\frac{1}{\phi}\left(g r_{i t}-\frac{A_{t-1}}{K_{i t}}\right) \equiv \bar{u}_{i t} . $$
$$ \frac{A_{i j}}{x_{i} x_{j}}=\frac{\tau_{i}}{x_{i}} \text { for any } i, j \in J_{x^{*}} $$ Similarly, $$ \frac{A_{r s}}{X_{r} \cdot X_{s}}=\frac{t_{s}}{X_{s}} \text { for any } r, s \in J_{\boldsymbol{x}} . $$ Since the indices $i$ and $s$ are in $J_{x}$, $$ A_{\text {is }}=t_{s} x_{i}=\tau_{i} x_{s} \text {, } $$ which in turn implies that $$ \frac{\tau_{i}}{x_{i}}=\frac{t_{s}}{x_{s}} . $$
John Maynard Keynes developed a new branch of economics called Keynesian economics, or more generally, macroeconomics. Keynes referred to the economists who came before him as “classical” economists and argued that while their theories might apply to individual choices and commodity markets, they did not adequately describe the workings of the economy as a whole, rather than marginal units or even specific commodity markets and prices. Keynesian macroeconomics describes economic unemployment, aggregate demand, or average price level inflation for all commodities in terms of massive aggregates representing the global economy. Keynesian theory suggests that the government can be a powerful player in the economy by implementing expansionary fiscal and monetary policies, manipulating government spending, taxation, and money creation to manage the economy in order to rescue it from recession.
宏观经济学课后作业代写
Let the random variable $S$ to follow a Pareto distribution with parameter $\alpha$. Thus, the probability distribution of $\log (S)$ is: $$ \operatorname{Pr}(\log (S) \geq k)=\operatorname{Pr}(S \geq \exp (k)) \propto(\exp (k))^{-\alpha}=\exp (-\alpha \mathrm{k}) $$ that is, an exponential distribution with parameter $\alpha$. In other terms, $\log (S)$ follows an exponential distribution with probability density function equal to: $$ E\left(\log (S) ; \alpha^{-1}\right)=\frac{1}{\alpha} \exp \left(-\frac{\log (S)}{\alpha}\right) $$ In the case of independent exponential variables, it is simple to prove that a Laplace distribution regarding growth rates emerges by making use of the convolution theorem and its relation with the characteristic function. In fact, the characteristic function of two independent exponential distributions $z_{j}, j=1,2$, with parameter $\alpha^{-1}$ is: $$ C_{z,}(\gamma)=(1-i \alpha \gamma)^{-1} $$
