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# 统计代写|时间序列分析代写Time Series Analysis代考|STAT435

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## 统计代写|时间序列分析代写Time Series Analysis代考|Trend Stationary Models

For the $T S$ approach we imagine the trend grows deterministically as:
$$T_t=A e^{\mu t}$$
where $\mu$ is the growth rate and $A$ is the value of $T_t$ at $t=0$.
Substituting $T_t=A e^{\mu t}$ into (1.3) we have:
$$W_t=A e^{\mu t+Y_t} .$$
so if we then define $X_t$ as the logarithm of $W_t$ as:
$$X_t \equiv \ln \left(W_t\right)$$
then we get the linear relationship:
$$X_t=\alpha+\mu t+Y_t$$
where $\alpha=\ln (A)$.
Since $Y_t$ the cycle will be assumed to be stationary, we see that $X_t$, is stationary except for the trend $\alpha+\mu t$, hence the terminology: trend stationary.
Suppose we have a sample of $T$ observations: $\left{W_1, W_2, \ldots W_T\right}$ of a particular time series. For TS models we can obtain estimates of $\alpha$ and $\mu$, the trend $T_t$ and the cycle $Y_t$ by applying ordinary least squares to (1.8). ${ }^1$
If the data are expressed as annual rates (i.e., GNP per year and not say $G N P$ per quarter) then for the vast majority of economic time series we would expect values of $\mu$ roughly in the range:
$$0 \leq \mu \leq 0.1$$
reflecting growth between zero and ten percent per year.
For example with post-war quarterly U.S. real consumption, which is plotted below, we might obtain:
$$X_t=\underset{(1479.58)}{6.48}+\underset{(196.1)}{0.0084 t}+Y_t$$
(the figures in brackets are $t$ statistics). If the data are expressed as consumption per quarter, the coefficient on time is a quarterly growth rate. To convert to an annual growth rate we multiply by 4 to obtain:
$$0.0084 \times 4=0.034$$
or an annual growth rate of about $3.4 \%$. Using the rule of 72 we would expect this series to double about every:
$$\frac{72}{3.4} \approx 20 \text { years. }$$

## 统计代写|时间序列分析代写Time Series Analysis代考|Difference Stationary Models

An alternative approach, made popular in the 1970’s by the work of Box and Jenkins, is the difference stationary (or $D S$ ) approach. Here we set
$$T_t=e^\mu W_{t-1},$$
that is as the previous period’s value of $W_t$ increased by $\mu \times 100 \%$ to reflect growth from period $t-1$ to $t$. Since $W_{t-1}$ is random, it follows that the trend $T_t$ is random. This is unlike the TS approach where the trend is nonrandom.. It is for this reason that people sometimes refer to $D S$ models as stochastic trend models.
Substituting (1.14) into (1.3) we obtain:
$$W_t=W_{t-1} e^{\mu+Y_t} .$$
Again if we define $X_t \equiv \ln \left(W_t\right)$ we have:
$$X_t=X_{t-1}+\mu+Y_t$$
or
$$\Delta X_t=\mu+Y_t .$$
where we use $\Delta$ to denote differences so that:
$$\Delta X_t \equiv X_t-X_{t-1}$$
Since $Y_t+\mu$ will turn out to be stationary, we see that $X_t$ is stationary once it is differenced, hence the terminology: difference stationary.
Now to obtain an estimate of the cycle $Y_t$ we regress $\Delta X_t$ on a constant. We might for example obtain:
$$\Delta X_t=\underset{(13.2)}{0.008}+Y_t$$
(where the figure in brackets is the $t$ statistic) and the implied annual growth rate:
$$0.008 \times 4=0.032$$
or $3.2 \%$ per year.
The cycle $Y_t$ for the $D S$ model can be obtained as the least squares residual from this regression or:
$$Y_t=\Delta X_t-0.008$$

## 统计代写|时间序列分析代写Time Series Analysis代考|Trend Stationary Models

$$T_t=A e^{\mu t}$$

$$W_t=A e^{\mu t+Y_t} .$$

$$X_t \equiv \ln \left(W_t\right)$$

$$X_t=\alpha+\mu t+Y_t$$

$$0 \leq \mu \leq 0.1$$

$$X_t=\underset{(1479.58)}{6.48}+\underset{(196.1)}{0.0084 t}+Y_t$$
(括号内的数字为$t$统计数字)。如果数据表示为每季度的消费，则时间系数是季度增长率。要换算成年增长率，我们将其乘以4得到:
$$0.0084 \times 4=0.034$$

$$\frac{72}{3.4} \approx 20 \text { years. }$$

## 统计代写|时间序列分析代写Time Series Analysis代考|Difference Stationary Models

$$T_t=e^\mu W_{t-1},$$

$$W_t=W_{t-1} e^{\mu+Y_t} .$$

$$X_t=X_{t-1}+\mu+Y_t$$

$$\Delta X_t=\mu+Y_t .$$

$$\Delta X_t \equiv X_t-X_{t-1}$$

$$\Delta X_t=\underset{(13.2)}{0.008}+Y_t$$
(括号内数字为$t$统计数字)及隐含的年增长率:
$$0.008 \times 4=0.032$$

$$Y_t=\Delta X_t-0.008$$

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