# 金融代写|金融实证代写Financial Empirical 代考|FINANCE630

## 金融代写|金融实证代写Financial Empirical 代考|Comparison of Forecasts

All the methods presented above will be used to fit model (1) as a forecasting model. For benchmarking, we will use the AR-4 model
$$\hat{y}t=\sum{i=1}^4 \beta_i y_{t-i}+\epsilon_t .$$
with four autoregressive terms and the more general AR-p model
$$\hat{y}t=\sum{i=1}^p \beta_i y_{t-i}+\epsilon_t, \quad j=1, \ldots, p_{\max }$$
where $p \leq p_{\max }$ is determined by AIC and $p_{\max }$ equals $q$ from (1). In $\mathrm{R}$, these models are calculated by the function ar from the stats package.

To measure the forecasting accuracy, the most popular choice is the relative mean squared forecasting error (RMSFE)
$$\operatorname{RMSFE}\left(\hat{y}t\right)=\frac{\sum{t=1}^T\left(\hat{y}t-y_t\right)^2}{\sum{t=1}^T\left(\hat{y}_{B t}-y_t\right)^2}$$
where $\hat{y}_t, t=1, \ldots, T$, is the predicted value for $y_t$. The RMSFE is calculated as the ratio of the mean squared forecasting errors of the corresponding method and the benchmark model $\hat{y}{B t}(\mathrm{AR}-\mathrm{p})$. Additionally, we will use the geometric mean relative absolute error (GMRAE) $$G M R A E\left(\hat{y}_t\right)=\left(\prod{t=1}^T \frac{\left|\hat{y}t-y_t\right|}{\left|\hat{y}{B t}-y_t\right|}\right)^{\frac{1}{T}}$$
as recommended by Armstrong and Collopy (1992). Here, corresponding forecasts are directly compared to each other and therefore the mean growth rate of the absolute forecasting error is compared to the benchmark forecast $\hat{y}B=\left(\hat{y}{B 1}, \ldots, \hat{y}_{B T}\right)$.

## 金融代写|金融实证代写Financial Empirical 代考|Ensemble Methods

The combination methods we use differ with regard to their complexity. Simple methods to combine a list of $n$ time series, $\hat{y}{1 t}, \ldots, \hat{y}{n t}$, for observation $t$ are the Arithmetic Mean
$$\hat{y}{c t}-\frac{1}{n} \sum{i=1}^n \hat{y}{t t},$$ the Median $$\hat{y}{c t}= \begin{cases}\hat{y}{\left(\frac{n+1}{2}\right) t}^2 & n \text { uneven } \ \frac{1}{2}\left(\hat{y}{\left(\frac{n}{2}\right) t}+\hat{y}{\left(\frac{n}{2}+1\right) t}\right) & n \text { even }\end{cases}$$ and the Trimmed Mean $$\hat{y}{c t}=\frac{1}{n-2 \cdot\lfloor\alpha n\rfloor} \sum_{i=\lfloor\alpha n\rfloor+1}^{n-\lfloor\alpha n\rfloor} \hat{y}{(i) t},$$ where $\hat{y}{(1) t}, \hat{y}{(2) t}, \ldots, \hat{y}{(n) t}$ represents the order statistic for observation $t$ and $\alpha$ represents the proportion of the highest and lowest forecasts eliminated in the trimmed mean. In our application, $\alpha=0.1$ is used.

The three methods do not use any information from former forecasts and can therefore also be used for rather small data sets. All the following methods try to use some information on the forecasting accuracy of the single methods in former forecasts.

## 金融代写|金融实证代写Financial Empirical 代考|Comparison of Forecasts

$$\operatorname{RMSFE}(\hat{y} t)=\frac{\sum t=1^T\left(\hat{y} t-y_t\right)^2}{\sum t=1^T\left(\hat{y}{B t}-y_t\right)^2}$$ 在哪里 $\hat{y}_t, t=1, \ldots, T$ ，是预测值 $y_t$. RMSFE 计算为相应方法的均方预测误差与基准模型的比值 $\hat{y} B t(\mathrm{AR}-\mathrm{p})$. 此外，我们将使用几何平均相对绝对误差 (GMRAE) $$G M R A E\left(\hat{y}_t\right)=\left(\prod t=1^T \frac{\left|\hat{y} t-y_t\right|}{\left|\hat{y} B t-y_t\right|}\right)^{\frac{1}{T}}$$ 根据 Armstrong 和 Collopy (1992) 的建议。在这里，相应的预测直接相互比较，因此绝对预测误差的平均 增长率与基准预测进行比较 $\hat{y} B=\left(\hat{y} B 1, \ldots, \hat{y}{B T}\right)$.

## 金融代写|金融实证代写Financial Empirical 代考|Ensemble Methods

$$\hat{y} c t-\frac{1}{n} \sum i=1^n \hat{y} t t,$$

$$\hat{y} c t=\left{\hat{y}\left(\frac{n+1}{2}\right) t^2 \quad n \text { uneven } \frac{1}{2}\left(\hat{y}\left(\frac{n}{2}\right) t+\hat{y}\left(\frac{n}{2}+1\right) t\right) \quad n\right. \text { even }$$

$$\hat{y} c t=\frac{1}{n-2 \cdot\lfloor\alpha n\rfloor} \sum_{i=\lfloor\alpha n\rfloor+1}^{n-\lfloor\alpha n\rfloor} \hat{y}(i) t,$$

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