### The Augmented Dickey-Fuller Test An Augmented Dickey-Fuller (ADF) test is a type of statistical test that determines whether a unit root is present in time series data. Unit roots can cause unpredictable results in time series analysis. A null hypothesis is formed in the unit root test to determine how strongly time series data is affected by a trend. By accepting the null hypothesis, we accept the evidence that the time series data is not stationary. By rejecting the null hypothesis or accepting the alternative hypothesis, we accept the evidence that the time series data is generated by a stationary process. This process is also known as stationary trend. The values of the ADF test statistic are negative. Lower ADF values indicate a stronger rejection of the null hypothesis. Augmented Dickey-Fuller Test is a common statistical test used to test whether a given time series is stationary or not. We can achieve this by defining the null and alternate hypothesis. Null Hypothesis: Time Series is non-stationary. It gives a time-dependent trend. Alternate Hypothesis: Time Series is stationary. In another term, the series doesn’t depend on time. ADF or t Statistic \< critical values: Reject the null hypothesis, time series is stationary. ADF or t Statistic > critical values: Failed to reject the null hypothesis, time series is non-stationary. Let’s check if our series that we are analyzing is a stationary series. Let’s create a function to check, using the `Dickey Fuller` test ```python theme={null} from statsmodels.tsa.stattools import adfuller def Augmented_Dickey_Fuller_Test_func(series , column_name): print (f'Dickey-Fuller test results for columns: {column_name}') dftest = adfuller(series, autolag='AIC') dfoutput = pd.Series(dftest[0:4], index=['Test Statistic','p-value','No Lags Used','Number of observations used']) for key,value in dftest[4].items(): dfoutput['Critical Value (%s)'%key] = value print (dfoutput) if dftest[1] <= 0.05: print("Conclusion:====>") print("Reject the null hypothesis") print("The data is stationary") else: print("Conclusion:====>") print("The null hypothesis cannot be rejected") print("The data is not stationary") ``` ```python theme={null} Augmented_Dickey_Fuller_Test_func(df["y"],'S&P500') ``` ```text theme={null} Dickey-Fuller test results for columns: S&P500 Test Statistic -0.814971 p-value 0.814685 No Lags Used 10.000000 ... Critical Value (1%) -3.433341 Critical Value (5%) -2.862861 Critical Value (10%) -2.567473 Length: 7, dtype: float64 Conclusion:====> The null hypothesis cannot be rejected The data is not stationary ``` In the previous result we can see that the `Augmented_Dickey_Fuller` test gives us a `p-value` of 0.864700, which tells us that the null hypothesis cannot be rejected, and on the other hand the data of our series are not stationary. We need to differentiate our time series, in order to convert the data to stationary. ### Return Series Since the 1970s, the financial industry has been very prosperous with advancement of computer and Internet technology. Trade of financial products (including various derivatives) generates a huge amount of data which form financial time series. For finance, the return on a financial product is most interesting, and so our attention focuses on the return series. If $P_t$ is the closing price at time t for a certain financial product, then the return on this product is $X_t = \frac{(P_t − P_{t−1})}{P_{t−1}} ≈ log(P_t ) − log(P_{t−1}).$ It is return series $\{X_t \}$ that have been much independently studied. And important stylized features which are common across many instruments, markets, and time periods have been summarized. Note that if you purchase the financial product, then it becomes your asset, and its returns become your asset returns. Now let us look at the following examples. We can estimate the series of returns using the [pandas](https://pandas.pydata.org/docs/reference/api/pandas.DataFrame.pct_change.html), `DataFrame.pct_change()` function. The `pct_change()` function has a periods parameter whose default value is 1. If you want to calculate a 30-day return, you must change the value to 30. ```python theme={null} df['return'] = 100 * df["y"].pct_change() df.dropna(inplace=True, how='any') df.head() ``` | | ds | y | unique\_id | return | | - | ------------------------- | ----------- | ---------- | --------- | | 1 | 2015-01-05 00:00:00+00:00 | 2020.579956 | 1 | -1.827811 | | 2 | 2015-01-06 00:00:00+00:00 | 2002.609985 | 1 | -0.889347 | | 3 | 2015-01-07 00:00:00+00:00 | 2025.900024 | 1 | 1.162984 | | 4 | 2015-01-08 00:00:00+00:00 | 2062.139893 | 1 | 1.788828 | | 5 | 2015-01-09 00:00:00+00:00 | 2044.810059 | 1 | -0.840381 | ```python theme={null} import plotly.express as px fig = px.line(df, x=df["ds"], y="return",title="SP500 Return Chart",template = "plotly_dark") fig.show() ```

### Creating Squared Returns ```python theme={null} df['sq_return'] = df["return"].mul(df["return"]) df.head() ``` | | ds | y | unique\_id | return | sq\_return | | - | ------------------------- | ----------- | ---------- | --------- | ---------- | | 1 | 2015-01-05 00:00:00+00:00 | 2020.579956 | 1 | -1.827811 | 3.340891 | | 2 | 2015-01-06 00:00:00+00:00 | 2002.609985 | 1 | -0.889347 | 0.790938 | | 3 | 2015-01-07 00:00:00+00:00 | 2025.900024 | 1 | 1.162984 | 1.352532 | | 4 | 2015-01-08 00:00:00+00:00 | 2062.139893 | 1 | 1.788828 | 3.199906 | | 5 | 2015-01-09 00:00:00+00:00 | 2044.810059 | 1 | -0.840381 | 0.706240 | ### Returns vs Squared Returns ```python theme={null} from plotly.subplots import make_subplots import plotly.graph_objects as go fig = make_subplots(rows=1, cols=2) fig.add_trace(go.Scatter(x=df["ds"], y=df["return"], mode='lines', name='return'), row=1, col=1 ) fig.add_trace(go.Scatter(x=df["ds"], y=df["sq_return"], mode='lines', name='sq_return'), row=1, col=2 ) fig.update_layout(height=600, width=800, title_text="Returns vs Squared Returns", template = "plotly_dark") fig.show() ```

```python theme={null} from scipy.stats import probplot, moment from statsmodels.tsa.stattools import adfuller, q_stat, acf import numpy as np import seaborn as sns def plot_correlogram(x, lags=None, title=None): lags = min(10, int(len(x)/5)) if lags is None else lags fig, axes = plt.subplots(nrows=2, ncols=2, figsize=(14, 8)) x.plot(ax=axes[0][0], title='Return') x.rolling(21).mean().plot(ax=axes[0][0], c='k', lw=1) q_p = np.max(q_stat(acf(x, nlags=lags), len(x))[1]) stats = f'Q-Stat: {np.max(q_p):>8.2f}\nADF: {adfuller(x)[1]:>11.2f}' axes[0][0].text(x=.02, y=.85, s=stats, transform=axes[0][0].transAxes) probplot(x, plot=axes[0][1]) mean, var, skew, kurtosis = moment(x, moment=[1, 2, 3, 4]) s = f'Mean: {mean:>12.2f}\nSD: {np.sqrt(var):>16.2f}\nSkew: {skew:12.2f}\nKurtosis:{kurtosis:9.2f}' axes[0][1].text(x=.02, y=.75, s=s, transform=axes[0][1].transAxes) plot_acf(x=x, lags=lags, zero=False, ax=axes[1][0]) plot_pacf(x, lags=lags, zero=False, ax=axes[1][1]) axes[1][0].set_xlabel('Lag') axes[1][1].set_xlabel('Lag') fig.suptitle(title+ f'Dickey-Fuller: {adfuller(x)[1]:>11.2f}', fontsize=14) sns.despine() fig.tight_layout() fig.subplots_adjust(top=.9) ``` ```python theme={null} plot_correlogram(df["return"], lags=30, title="Time Series Analysis plot \n") ```

### Ljung-Box Test Ljung-Box is a test for autocorrelation that we can use in tandem with our ACF and PACF plots. The Ljung-Box test takes our data, optionally either lag values to test, or the largest lag value to consider, and whether to compute the Box-Pierce statistic. Ljung-Box and Box-Pierce are two similar test statisitcs, Q , that are compared against a chi-squared distribution to determine if the series is white noise. We might use the Ljung-Box test on the residuals of our model to look for autocorrelation, ideally our residuals would be white noise. * Ho : The data are independently distributed, no autocorrelation. * Ha : The data are not independently distributed; they exhibit serial correlation. The Ljung-Box with the Box-Pierce option will return, for each lag, the Ljung-Box test statistic, Ljung-Box p-values, Box-Pierce test statistic, and Box-Pierce p-values. If $p<\alpha (0.05)$ we reject the null hypothesis. ```python theme={null} from statsmodels.stats.diagnostic import acorr_ljungbox ljung_res = acorr_ljungbox(df["return"], lags= 40, boxpierce=True) ljung_res.head() ``` | | lb\_stat | lb\_pvalue | bp\_stat | bp\_pvalue | | - | --------- | ------------ | --------- | ------------ | | 1 | 49.222273 | 2.285409e-12 | 49.155183 | 2.364927e-12 | | 2 | 62.991348 | 2.097020e-14 | 62.899234 | 2.195861e-14 | | 3 | 63.944944 | 8.433622e-14 | 63.850663 | 8.834380e-14 | | 4 | 74.343652 | 2.742989e-15 | 74.221024 | 2.911751e-15 | | 5 | 80.234862 | 7.494100e-16 | 80.093498 | 8.022242e-16 | ## Split the data into training and testing Let’s divide our data into sets 1. Data to train our `ARCH` model 2. Data to test our model For the test data we will use the last 30 day to test and evaluate the performance of our model. ```python theme={null} df=df[["ds","unique_id","return"]] df.columns=["ds", "unique_id", "y"] ``` ```python theme={null} train = df[df.ds<='2023-05-24'] # Let's forecast the last 30 days test = df[df.ds>'2023-05-24'] ``` ```python theme={null} train.shape, test.shape ``` ```text theme={null} ((2112, 3), (82, 3)) ``` Now let’s plot the training data and the test data. ```python theme={null} sns.lineplot(train,x="ds", y="y", label="Train") sns.lineplot(test, x="ds", y="y", label="Test") plt.show() ```

## Implementation of ARCH with StatsForecast To also know more about the parameters of the functions of the `ARCH Model`, they are listed below. For more information, visit the [documentation](../../src/core/models.html#arch) ```text theme={null} p : int Number of lagged versions of the series. alias : str Custom name of the model. prediction_intervals : Optional[ConformalIntervals] Information to compute conformal prediction intervals. By default, the model will compute the native prediction intervals. ``` ### Load libraries ```python theme={null} from statsforecast import StatsForecast from statsforecast.models import ARCH ``` ### Building Model Import and instantiate the models. Setting the argument is sometimes tricky. This article on [Seasonal periods](https://robjhyndman.com/hyndsight/seasonal-periods/) by the master, Rob Hyndmann, can be useful.season\_length. ```python theme={null} season_length = 7 # Daily data horizon = len(test) # number of predictions biasadj=True, include_drift=True, models = [ARCH(p=2)] ``` We fit the models by instantiating a new StatsForecast object with the following parameters: models: a list of models. Select the models you want from models and import them. * `freq:` a string indicating the frequency of the data. (See [pandas’ available frequencies](https://pandas.pydata.org/pandas-docs/stable/user_guide/timeseries.html#offset-aliases).) * `n_jobs:` n\_jobs: int, number of jobs used in the parallel processing, use -1 for all cores. * `fallback_model:` a model to be used if a model fails. Any settings are passed into the constructor. Then you call its fit method and pass in the historical data frame. ```python theme={null} sf = StatsForecast(models=models, freq='C', # custom business day frequency ) ``` ### Fit the Model ```python theme={null} sf.fit(df=train) ``` ```text theme={null} StatsForecast(models=[ARCH(2)]) ``` Let’s see the results of our ARCH model. We can observe it with the following instruction: ```python theme={null} result=sf.fitted_[0,0].model_ result ``` ```text theme={null} {'p': 2, 'q': 0, 'coeff': array([0.44321058, 0.34706751, 0.35172097]), 'message': 'Optimization terminated successfully', 'y_vals': array([-1.12220267, -0.73186003]), 'sigma2_vals': array([1.38768694, nan, 1.89278112, ..., 0.76423271, 0.45064684, 0.88037072]), 'fitted': array([ nan, nan, 2.23474807, ..., -1.48033228, 1.10018999, -0.98050166]), 'actual_residuals': array([ nan, nan, -1.07176381, ..., 1.49583575, -2.22239266, 0.24864162])} ``` Let us now visualize the residuals of our models. As we can see, the result obtained above has an output in a dictionary, to extract each element from the dictionary we are going to use the `.get()` function to extract the element and then we are going to save it in a `pd.DataFrame()`. ```python theme={null} residual=pd.DataFrame(result.get("actual_residuals"), columns=["residual Model"]) residual ``` | | residual Model | | ---- | -------------- | | 0 | NaN | | 1 | NaN | | 2 | -1.071764 | | ... | ... | | 2109 | 1.495836 | | 2110 | -2.222393 | | 2111 | 0.248642 | ```python theme={null} import scipy.stats as stats fig, axs = plt.subplots(nrows=2, ncols=2) # plot[1,1] residual.plot(ax=axs[0,0]) axs[0,0].set_title("Residuals"); # plot sns.distplot(residual, ax=axs[0,1]); axs[0,1].set_title("Density plot - Residual"); # plot stats.probplot(residual["residual Model"], dist="norm", plot=axs[1,0]) axs[1,0].set_title('Plot Q-Q') # plot plot_acf(residual, lags=35, ax=axs[1,1],color="fuchsia") axs[1,1].set_title("Autocorrelation"); plt.show(); ```

### Forecast Method If you want to gain speed in productive settings where you have multiple series or models we recommend using the `StatsForecast.forecast` method instead of `.fit` and `.predict`. The main difference is that the `.forecast` doest not store the fitted values and is highly scalable in distributed environments. The forecast method takes two arguments: forecasts next `h` (horizon) and `level`. * `h (int):` represents the forecast h steps into the future. In this case, 12 months ahead. * `level (list of floats):` this optional parameter is used for probabilistic forecasting. Set the level (or confidence percentile) of your prediction interval. For example, `level=[90]` means that the model expects the real value to be inside that interval 90% of the times. The forecast object here is a new data frame that includes a column with the name of the model and the y hat values, as well as columns for the uncertainty intervals. Depending on your computer, this step should take around 1min. ```python theme={null} Y_hat = sf.forecast(df=train, h=horizon, fitted=True) Y_hat ``` | | unique\_id | ds | ARCH(2) | | --- | ---------- | ------------------------- | --------- | | 0 | 1 | 2023-05-25 00:00:00+00:00 | 1.681839 | | 1 | 1 | 2023-05-26 00:00:00+00:00 | -0.777029 | | 2 | 1 | 2023-05-29 00:00:00+00:00 | -0.677962 | | ... | ... | ... | ... | | 79 | 1 | 2023-09-13 00:00:00+00:00 | 0.695591 | | 80 | 1 | 2023-09-14 00:00:00+00:00 | -0.176075 | | 81 | 1 | 2023-09-15 00:00:00+00:00 | -0.158605 | ```python theme={null} values=sf.forecast_fitted_values() values.head() ``` | | unique\_id | ds | y | ARCH(2) | | - | ---------- | ------------------------- | --------- | --------- | | 0 | 1 | 2015-01-05 00:00:00+00:00 | -1.827811 | NaN | | 1 | 1 | 2015-01-06 00:00:00+00:00 | -0.889347 | NaN | | 2 | 1 | 2015-01-07 00:00:00+00:00 | 1.162984 | 2.234748 | | 3 | 1 | 2015-01-08 00:00:00+00:00 | 1.788828 | -0.667577 | | 4 | 1 | 2015-01-09 00:00:00+00:00 | -0.840381 | -0.752438 | Adding 95% confidence interval with the forecast method ```python theme={null} sf.forecast(df=train, h=horizon, level=[95]) ``` | | unique\_id | ds | ARCH(2) | ARCH(2)-lo-95 | ARCH(2)-hi-95 | | --- | ---------- | ------------------------- | --------- | ------------- | ------------- | | 0 | 1 | 2023-05-25 00:00:00+00:00 | 1.681839 | -0.419326 | 3.783003 | | 1 | 1 | 2023-05-26 00:00:00+00:00 | -0.777029 | -3.939054 | 2.384996 | | 2 | 1 | 2023-05-29 00:00:00+00:00 | -0.677962 | -3.907262 | 2.551338 | | ... | ... | ... | ... | ... | ... | | 79 | 1 | 2023-09-13 00:00:00+00:00 | 0.695591 | -0.937585 | 2.328766 | | 80 | 1 | 2023-09-14 00:00:00+00:00 | -0.176075 | -1.405359 | 1.053210 | | 81 | 1 | 2023-09-15 00:00:00+00:00 | -0.158605 | -1.381915 | 1.064705 | ```python theme={null} # Merge the forecasts with the true values Y_hat1 = test.merge(Y_hat, how='left', on=['unique_id', 'ds']) Y_hat1 ``` | | ds | unique\_id | y | ARCH(2) | | --- | ------------------------- | ---------- | --------- | --------- | | 0 | 2023-05-25 00:00:00+00:00 | 1 | 0.875758 | 1.681839 | | 1 | 2023-05-26 00:00:00+00:00 | 1 | 1.304909 | -0.777029 | | 2 | 2023-05-30 00:00:00+00:00 | 1 | 0.001660 | -0.968703 | | ... | ... | ... | ... | ... | | 79 | 2023-09-19 00:00:00+00:00 | 1 | -0.215101 | NaN | | 80 | 2023-09-20 00:00:00+00:00 | 1 | -0.939479 | NaN | | 81 | 2023-09-21 00:00:00+00:00 | 1 | -1.640093 | NaN | ```python theme={null} # Merge the forecasts with the true values fig, ax = plt.subplots(1, 1) plot_df = pd.concat([train, Y_hat1]).set_index('ds') plot_df[['y', "ARCH(2)"]].plot(ax=ax, linewidth=2) ax.set_title(' Forecast', fontsize=22) ax.set_ylabel('Year ', fontsize=20) ax.set_xlabel('Timestamp [t]', fontsize=20) ax.legend(prop={'size': 15}) ax.grid(True) plt.show() ```

### Predict method with confidence interval To generate forecasts use the predict method. The predict method takes two arguments: forecasts the next `h` (for horizon) and `level`. * `h (int):` represents the forecast h steps into the future. In this case, 12 months ahead. * `level (list of floats):` this optional parameter is used for probabilistic forecasting. Set the level (or confidence percentile) of your prediction interval. For example, `level=[95]` means that the model expects the real value to be inside that interval 95% of the times. The forecast object here is a new data frame that includes a column with the name of the model and the y hat values, as well as columns for the uncertainty intervals. This step should take less than 1 second. ```python theme={null} sf.predict(h=horizon) ``` | | unique\_id | ds | ARCH(2) | | --- | ---------- | ------------------------- | --------- | | 0 | 1 | 2023-05-25 00:00:00+00:00 | 1.681839 | | 1 | 1 | 2023-05-26 00:00:00+00:00 | -0.777029 | | 2 | 1 | 2023-05-29 00:00:00+00:00 | -0.677962 | | ... | ... | ... | ... | | 79 | 1 | 2023-09-13 00:00:00+00:00 | 0.695591 | | 80 | 1 | 2023-09-14 00:00:00+00:00 | -0.176075 | | 81 | 1 | 2023-09-15 00:00:00+00:00 | -0.158605 | ```python theme={null} forecast_df = sf.predict(h=horizon, level=[80,95]) forecast_df ``` | | unique\_id | ds | ARCH(2) | ARCH(2)-lo-95 | ARCH(2)-lo-80 | ARCH(2)-hi-80 | ARCH(2)-hi-95 | | --- | ---------- | ------------------------- | --------- | ------------- | ------------- | ------------- | ------------- | | 0 | 1 | 2023-05-25 00:00:00+00:00 | 1.681839 | -0.419326 | 0.307961 | 3.055716 | 3.783003 | | 1 | 1 | 2023-05-26 00:00:00+00:00 | -0.777029 | -3.939054 | -2.844566 | 1.290508 | 2.384996 | | 2 | 1 | 2023-05-29 00:00:00+00:00 | -0.677962 | -3.907262 | -2.789488 | 1.433564 | 2.551338 | | ... | ... | ... | ... | ... | ... | ... | ... | | 79 | 1 | 2023-09-13 00:00:00+00:00 | 0.695591 | -0.937585 | -0.372285 | 1.763467 | 2.328766 | | 80 | 1 | 2023-09-14 00:00:00+00:00 | -0.176075 | -1.405359 | -0.979860 | 0.627711 | 1.053210 | | 81 | 1 | 2023-09-15 00:00:00+00:00 | -0.158605 | -1.381915 | -0.958485 | 0.641274 | 1.064705 | We can join the forecast result with the historical data using the pandas function `pd.concat()`, and then be able to use this result for graphing. ```python theme={null} df_plot=pd.concat([df, forecast_df]).set_index('ds').tail(220) df_plot ``` | | unique\_id | y | ARCH(2) | ARCH(2)-lo-95 | ARCH(2)-lo-80 | ARCH(2)-hi-80 | ARCH(2)-hi-95 | | ------------------------- | ---------- | --------- | --------- | ------------- | ------------- | ------------- | ------------- | | ds | | | | | | | | | 2023-03-07 00:00:00+00:00 | 1 | -1.532692 | NaN | NaN | NaN | NaN | NaN | | 2023-03-08 00:00:00+00:00 | 1 | 0.141479 | NaN | NaN | NaN | NaN | NaN | | 2023-03-09 00:00:00+00:00 | 1 | -1.845936 | NaN | NaN | NaN | NaN | NaN | | ... | ... | ... | ... | ... | ... | ... | ... | | 2023-09-13 00:00:00+00:00 | 1 | NaN | 0.695591 | -0.937585 | -0.372285 | 1.763467 | 2.328766 | | 2023-09-14 00:00:00+00:00 | 1 | NaN | -0.176075 | -1.405359 | -0.979860 | 0.627711 | 1.053210 | | 2023-09-15 00:00:00+00:00 | 1 | NaN | -0.158605 | -1.381915 | -0.958485 | 0.641274 | 1.064705 | ```python theme={null} sf.plot(train, test.merge(forecast_df), level=[80, 95], max_insample_length=120) ```

Let’s plot the same graph using the plot function that comes in `Statsforecast`, as shown below. ## Cross-validation In previous steps, we’ve taken our historical data to predict the future. However, to asses its accuracy we would also like to know how the model would have performed in the past. To assess the accuracy and robustness of your models on your data perform Cross-Validation. With time series data, Cross Validation is done by defining a sliding window across the historical data and predicting the period following it. This form of cross-validation allows us to arrive at a better estimation of our model’s predictive abilities across a wider range of temporal instances while also keeping the data in the training set contiguous as is required by our models. The following graph depicts such a Cross Validation Strategy: ![](https://raw.githubusercontent.com/Nixtla/statsforecast/main/nbs/imgs/ChainedWindows.gif) ### Perform time series cross-validation Cross-validation of time series models is considered a best practice but most implementations are very slow. The statsforecast library implements cross-validation as a distributed operation, making the process less time-consuming to perform. If you have big datasets you can also perform Cross Validation in a distributed cluster using Ray, Dask or Spark. In this case, we want to evaluate the performance of each model for the last 5 months `(n_windows=5)`, forecasting every second months `(step_size=12)`. Depending on your computer, this step should take around 1 min. The cross\_validation method from the StatsForecast class takes the following arguments. * `df:` training data frame * `h (int):` represents h steps into the future that are being forecasted. In this case, 12 months ahead. * `step_size (int):` step size between each window. In other words: how often do you want to run the forecasting processes. * `n_windows(int):` number of windows used for cross validation. In other words: what number of forecasting processes in the past do you want to evaluate. ```python theme={null} crossvalidation_df = sf.cross_validation(df=train, h=horizon, step_size=6, n_windows=5) ``` The crossvaldation\_df object is a new data frame that includes the following columns: * `unique_id:` series identifier * `ds:` datestamp or temporal index * `cutoff:` the last datestamp or temporal index for the n\_windows. * `y:` true value * `"model":` columns with the model’s name and fitted value. ```python theme={null} crossvalidation_df ``` | | unique\_id | ds | cutoff | y | ARCH(2) | | --- | ---------- | ------------------------- | ------------------------- | --------- | --------- | | 0 | 1 | 2022-12-21 00:00:00+00:00 | 2022-12-20 00:00:00+00:00 | 1.486799 | 1.382105 | | 1 | 1 | 2022-12-22 00:00:00+00:00 | 2022-12-20 00:00:00+00:00 | -1.445170 | -0.651618 | | 2 | 1 | 2022-12-23 00:00:00+00:00 | 2022-12-20 00:00:00+00:00 | 0.586810 | -0.595213 | | ... | ... | ... | ... | ... | ... | | 407 | 1 | 2023-05-22 00:00:00+00:00 | 2023-01-26 00:00:00+00:00 | 0.015503 | 0.693070 | | 408 | 1 | 2023-05-23 00:00:00+00:00 | 2023-01-26 00:00:00+00:00 | -1.122203 | -0.176181 | | 409 | 1 | 2023-05-24 00:00:00+00:00 | 2023-01-26 00:00:00+00:00 | -0.731860 | -0.157522 | ## Model Evaluation Now we are going to evaluate our model with the results of the predictions, we will use different types of metrics MAE, MAPE, MASE, RMSE, SMAPE to evaluate the accuracy. ```python theme={null} from functools import partial from utilsforecast.evaluation import evaluate from utilsforecast.losses import mae, mape, mase, rmse, smape ``` ```python theme={null} evaluate( test.merge(Y_hat), train_df=train, metrics=[mae, mape, partial(mase, seasonality=5), rmse, smape], agg_fn='mean', ) ``` | | metric | ARCH(2) | | - | ------ | --------- | | 0 | mae | 0.949721 | | 1 | mape | 11.789856 | | 2 | mase | 0.875298 | | 3 | rmse | 1.164914 | | 4 | smape | 0.725702 | ## References 1. [Changquan Huang • Alla Petukhina. Springer series (2022). Applied Time Series Analysis and Forecasting with Python.](https://link.springer.com/book/10.1007/978-3-031-13584-2) 2. [Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: Journal of the econometric society, 987-1007.](http://www.econ.uiuc.edu/~econ508/Papers/engle82.pdf). 3. [James D. Hamilton. Time Series Analysis Princeton University Press, Princeton, New Jersey, 1st Edition, 1994.](https://press.princeton.edu/books/hardcover/9780691042893/time-series-analysis) 4. [Nixtla ARCH API](../../src/core/models.html#arch) 5. [Pandas available frequencies](https://pandas.pydata.org/pandas-docs/stable/user_guide/timeseries.html#offset-aliases). 6. [Rob J. Hyndman and George Athanasopoulos (2018). “Forecasting Principles and Practice (3rd ed)”](https://otexts.com/fpp3/tscv.html). 7. [Seasonal periods- Rob J Hyndman](https://robjhyndman.com/hyndsight/seasonal-periods/).