## ARIMA Model Results ### Overview The image presents the results of an ARIMA (Autoregressive Integrated Moving Average) model. It includes model specifications, goodness-of-fit statistics, coefficient estimates, and root analysis. ### Components/Axes **Header:** * Title: ARIMA Model Results * Dep. Variable: D.Close * Model: ARIMA(4, 1, 0) * Method: css-mle * Date: Mon, 02 May 2022 * Time: 03:16:01 * Sample: 05-03-2021 - 02-17-2022 * No. Observations: 291 * Log Likelihood: -2586.480 * S.D. of innovations: 1753.258 * AIC: 5184.959 * BIC: 5206.999 * HQIC: 5193.789 **Coefficient Estimates Table:** * Columns: coef, std err, z, P>|z|, [0.025, 0.975] * Rows: const, ar.L1.D.Close, ar.L2.D.Close, ar.L3.D.Close, ar.L4.D.Close **Roots Table:** * Columns: Real, Imaginary, Modulus, Frequency * Rows: AR.1, AR.2, AR.3, AR.4 ### Detailed Analysis **Coefficient Estimates Table:** | | coef | std err | z | P>\|z\| | \[0.025 | 0.975] | |---------------|-----------|---------|---------|---------|-----------|-----------| | const | -55.2311 | 101.025 | -0.547 | 0.585 | -253.236 | 142.774 | | ar.L1.D.Close | -0.0541 | 0.059 | -0.923 | 0.357 | -0.169 | 0.061 | | ar.L2.D.Close | -0.0278 | 0.059 | -0.470 | 0.638 | -0.143 | 0.088 | | ar.L3.D.Close | -0.0297 | 0.060 | -0.499 | 0.618 | -0.147 | 0.087 | | ar.L4.D.Close | 0.0947 | 0.060 | 1.592 | 0.113 | -0.022 | 0.211 | **Roots Table:** | | Real | Imaginary | Modulus | Frequency | |-------|-----------|-----------|---------|-----------| | AR.1 | -1.7241 | -0.0000j | 1.7241 | -0.5000 | | AR.2 | 0.0309 | -1.7600j | 1.7603 | -0.2472 | | AR.3 | 0.0309 | +1.7600j | 1.7603 | 0.2472 | | AR.4 | 1.9763 | -0.0000j | 1.9763 | -0.0000 | ### Key Observations * The dependent variable is "D.Close". * The ARIMA model is of order (4, 1, 0). * The sample period is from 05-03-2021 to 02-17-2022. * Most of the coefficients for the autoregressive terms (ar.L1.D.Close to ar.L4.D.Close) are not statistically significant at a 5% significance level (P>|z| > 0.05). * The roots of the AR polynomial have varying real and imaginary components, with corresponding moduli and frequencies. ### Interpretation The ARIMA model results provide insights into the time series behavior of "D.Close". The model attempts to capture the autocorrelation structure in the data using four lagged terms. However, the high p-values for most of the autoregressive coefficients suggest that these lags may not be strong predictors of the current value of "D.Close". The roots analysis provides information about the stability and oscillatory behavior of the time series. The moduli of the roots are all greater than 1, suggesting that the model is stationary. The frequencies indicate the cyclical patterns present in the data.

\n ## Statistical Output: ARIMA Model Results ### Overview This image presents the results of an ARIMA (Autoregressive Integrated Moving Average) model applied to time series data. The output details model parameters, statistical tests, and root analysis. The data appears to be financial, specifically "D.Close" (presumably Daily Close price). ### Components/Axes The output is structured into several sections: * **Model Information:** Includes dependent variable, model specification (ARIMA(4,1,0)), method (css-mle), date, time, and sample period. * **Model Fit Statistics:** Provides metrics like No. Observations, Log Likelihood, S.D. of innovations, AIC, BIC, and HQIC. * **Parameter Estimates:** A table displaying coefficients (coef), standard errors (std err), z-statistics (z), p-values (P>|z|), and confidence intervals ([0.025, 0.975]) for each model parameter. * **Root Analysis:** A table showing the real, imaginary, modulus, and frequency components of the roots of the characteristic polynomial. ### Detailed Analysis or Content Details **Model Information:** * Dependent Variable: D.Close * Model: ARIMA(4, 1, 0) * Method: css-mle * Date: Mon, 02 May 2022 * Time: 03:16:01 * Sample: 05-03-2021 - 02-17-2022 * No. Observations: 291 * Log Likelihood: -2586.480 * S.D. of innovations: 1753.258 * AIC: 5184.959 * BIC: 5206.999 * HQIC: 5193.789 **Parameter Estimates:** | Parameter | coef | std err | z | P>|z| | [0.025 | 0.975 | |-----------|---------|---------|--------|-------|-------|----------|----------| | const | -55.2311| 101.025 | -0.547 | 0.585 | -253.236| 142.774 | | ar.L1.D.Close | -0.0541 | 0.059 | -0.923 | 0.357 | -0.169 | 0.061 | | ar.L2.D.Close | -0.0278 | 0.059 | -0.470 | 0.638 | -0.143 | 0.088 | | ar.L3.D.Close | -0.0297 | 0.060 | -0.499 | 0.618 | -0.147 | 0.087 | | ar.L4.D.Close | 0.0947 | 0.060 | 1.592 | 0.113 | -0.022 | 0.211 | **Root Analysis:** | Root | Real | Imaginary | Modulus | Frequency | |-------|---------|-----------|---------|-----------| | AR.1 | -1.7241 | -0.0000j | 1.7241 | -0.5000 | | AR.2 | 0.0309 | -1.7600j | 1.7603 | -0.2472 | | AR.3 | 0.0309 | +1.7600j | 1.7603 | 0.2472 | | AR.4 | 1.9763 | -0.0000j | 1.9763 | -0.0000 | *Note: 'j' denotes the imaginary unit.* ### Key Observations * The p-values for all AR coefficients are relatively high (greater than 0.05), suggesting that these coefficients are not statistically significant at the 5% level. * The constant term also has a high p-value, indicating it's not statistically significant. * The roots of the characteristic polynomial have moduli greater than 1, indicating that the ARIMA model is stationary. * The imaginary components of AR.2 and AR.3 roots are non-zero, indicating oscillatory behavior. ### Interpretation The ARIMA(4,1,0) model was fitted to the daily closing price (D.Close) time series data from March 5, 2021, to February 17, 2022. The model attempts to capture the autocorrelation structure in the data using four autoregressive (AR) terms. However, the statistical significance of the estimated coefficients is questionable, as indicated by the high p-values. This suggests that the model may not be a good fit for the data, or that a simpler model might be sufficient. The root analysis confirms the stationarity of the model, which is a necessary condition for valid inference. The presence of complex roots (AR.2 and AR.3) indicates that the time series exhibits some oscillatory behavior. The relatively large standard errors of the coefficients suggest that the estimates are imprecise, potentially due to a limited sample size or high noise in the data. The AIC, BIC, and HQIC values provide measures of model fit, but their interpretation requires comparison with other potential models. The negative log-likelihood suggests the model is attempting to fit the data, but the high AIC/BIC/HQIC values suggest it may not be doing so efficiently.

## Statistical Output: ARIMA Model Results ### Overview The image displays the textual output of a statistical model fitting, specifically an ARIMA (Autoregressive Integrated Moving Average) model. The output is presented in a monospaced font, typical of console or log output from statistical software. It contains three distinct sections: a header with model metadata, a table of estimated coefficients with their statistics, and a table of the model's autoregressive (AR) roots. ### Components/Axes The output is structured into three main components: 1. **Header Section:** Contains metadata about the model and the data. * **Dep. Variable:** `D.Close` (The dependent variable is the differenced closing price). * **Model:** `ARIMA(4, 1, 0)` (An ARIMA model with 4 autoregressive terms, 1 degree of differencing, and 0 moving average terms). * **Method:** `css-mle` (Conditional Sum of Squares - Maximum Likelihood Estimation). * **Date/Time:** `Mon, 02 May 2022`, `03:16:01`. * **Sample:** `05-03-2021 - 02-17-2022` (The date range of the data used for fitting). * **No. Observations:** `291`. * **Log Likelihood:** `-2586.480`. * **S.D. of innovations:** `1753.258`. * **AIC:** `5184.959` (Akaike Information Criterion). * **BIC:** `5206.999` (Bayesian Information Criterion). * **HQIC:** `5193.789` (Hannan-Quinn Information Criterion). 2. **Coefficients Table:** A table with 7 columns and 5 rows (including the header row). * **Columns:** `coef`, `std err`, `z`, `P>|z|`, `[0.025`, `0.975]`. * **Rows (Variables):** * `const`: Constant term. * `ar.L1.D.Close`: First lag of the differenced dependent variable. * `ar.L2.D.Close`: Second lag. * `ar.L3.D.Close`: Third lag. * `ar.L4.D.Close`: Fourth lag. 3. **Roots Table:** A table describing the properties of the autoregressive polynomial's roots. * **Columns:** `Real`, `Imaginary`, `Modulus`, `Frequency`. * **Rows (Roots):** `AR.1`, `AR.2`, `AR.3`, `AR.4`. ### Detailed Analysis **Coefficients Table Data:** | Variable | coef | std err | z | P>\|z\| | [0.025 | 0.975] | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | const | -55.2311 | 101.025 | -0.547 | 0.585 | -253.236 | 142.774 | | ar.L1.D.Close | -0.0541 | 0.059 | -0.923 | 0.357 | -0.169 | 0.061 | | ar.L2.D.Close | -0.0278 | 0.059 | -0.470 | 0.638 | -0.143 | 0.088 | | ar.L3.D.Close | -0.0297 | 0.060 | -0.499 | 0.618 | -0.147 | 0.087 | | ar.L4.D.Close | 0.0947 | 0.060 | 1.592 | 0.113 | -0.022 | 0.211 | **Roots Table Data:** | Root | Real | Imaginary | Modulus | Frequency | | :--- | :--- | :--- | :--- | :--- | | AR.1 | -1.7241 | -0.0000j | 1.7241 | -0.5000 | | AR.2 | 0.0309 | -1.7600j | 1.7603 | -0.2472 | | AR.3 | 0.0309 | +1.7600j | 1.7603 | 0.2472 | | AR.4 | 1.9763 | -0.0000j | 1.9763 | -0.0000 | ### Key Observations 1. **Model Specification:** The model is an ARIMA(4,1,0), meaning it uses four past values of the *differenced* series (`D.Close`) to predict its next value, with no moving average component. 2. **Coefficient Significance:** None of the estimated coefficients (including the constant) are statistically significant at the conventional 5% level (`P > |z| > 0.05`). The highest absolute z-statistic is 1.592 for the fourth lag (`ar.L4.D.Close`), with a p-value of 0.113. 3. **Roots and Stationarity:** All four AR roots have a modulus greater than 1 (ranging from 1.7241 to 1.9763). This indicates the AR polynomial is stationary, which is a required property for a valid ARIMA model. The presence of complex roots (AR.2 and AR.3) suggests potential cyclical behavior in the model's dynamics. 4. **Data Range:** The model was fitted on 291 observations of daily data (implied by the date format) spanning from May 3, 2021, to February 17, 2022. 5. **Fit Statistics:** The AIC, BIC, and HQIC values are provided for model comparison. The standard deviation of the innovations (residuals) is approximately 1753.26, indicating the scale of the model's prediction errors. ### Interpretation This output represents the results of fitting a linear time series model to financial data (likely a stock or index closing price, given the variable name `Close`). The primary finding is that a simple ARIMA(4,1,0) model does not find strong, statistically significant autoregressive patterns in the differenced data over this specific period. The lack of significant coefficients suggests that the past four days' price changes do not have a reliable linear relationship with the next day's change, according to this model. The stationarity of the roots confirms the model's mathematical validity but does not imply good predictive power. The high p-values for all coefficients indicate that this specific model structure may be overfit or that the underlying data is largely unpredictable using this linear method. A practitioner would likely use these results to conclude that this ARIMA specification is not useful for forecasting and would experiment with different lag orders, include moving average terms, or consider alternative, possibly non-linear, models. The large standard error of the constant term and the wide confidence intervals for all coefficients further underscore the uncertainty in these estimates.

## ARIMA Model Results: ARIMA(4, 1, 0) for D.Close ### Overview The image presents the results of an ARIMA(4, 1, 0) model applied to the dependent variable "D.Close" (likely a time series of closing prices). The analysis includes model parameters, statistical metrics, and diagnostic outputs. The data spans 291 observations, with a sample period from 05-03-2021 to 02-17-2022. ### Components/Axes - **Model Specification**: - Dependent Variable: `D.Close` - Model: `ARIMA(4, 1, 0)` - Method: `css-mle` (conditional sum of squares with maximum likelihood estimation) - Number of Observations: `291` - Log Likelihood: `-2586.480` - Standard Deviation of Innovations: `1753.258` - AIC: `5184.959` - BIC: `5206.999` - HQIC: `5193.789` - Date: `Mon, 02 May 2022` - Time: `03:16:01` - Sample Period: `05-03-2021` to `02-17-2022` - **Coefficients Table**: - **Columns**: `coef` (coefficient), `std err` (standard error), `z` (z-score), `P>|z|` (p-value), `[0.025` (lower 95% CI), `[0.975` (upper 95% CI) - **Rows**: - `const`: `-55.2311` (std err: `101.025`, z: `-0.547`, p: `0.585`) - `ar.L1.D.Close`: `-0.0541` (std err: `0.059`, z: `-0.923`, p: `0.357`) - `ar.L2.D.Close`: `-0.0278` (std err: `0.059`, z: `-0.470`, p: `0.638`) - `ar.L3.D.Close`: `-0.0297` (std err: `0.060`, z: `-0.499`, p: `0.618`) - `ar.L4.D.Close`: `0.0947` (std err: `0.060`, z: `1.592`, p: `0.113`) - **Roots Table**: - **Columns**: `Real`, `Imaginary`, `Modulus`, `Frequency` - **Rows**: - `AR.1`: Real: `-1.7241`, Imaginary: `-0.0000j`, Modulus: `1.7241`, Frequency: `-0.5000` - `AR.2`: Real: `0.0309`, Imaginary: `-1.7600j`, Modulus: `1.7603`, Frequency: `-0.2472` - `AR.3`: Real: `0.0309`, Imaginary: `+1.7600j`, Modulus: `1.7603`, Frequency: `0.2472` - `AR.4`: Real: `1.9763`, Imaginary: `-0.0000j`, Modulus: `1.9763`, Frequency: `0.0000` ### Detailed Analysis 1. **Model Parameters**: - The ARIMA(4,1,0) model includes 4 autoregressive terms (AR.1–AR.4), 1 differencing step (d=1), and no moving average terms (q=0). - The constant term (`const`) has a coefficient of `-55.2311`, but its p-value (`0.585`) suggests it is not statistically significant. 2. **Autoregressive Coefficients**: - **AR.1**: `-0.0541` (p=0.357) – non-significant. - **AR.2**: `-0.0278` (p=0.638) – non-significant. - **AR.3**: `-0.0297` (p=0.618) – non-significant. - **AR.4**: `0.0947` (p=0.113) – marginally significant (close to 0.05). 3. **Roots Analysis**: - **AR.1**: Root modulus = `1.7241` (outside the unit circle, indicating potential instability). - **AR.2/AR.3**: Complex conjugate roots with modulus `1.7603` (also outside the unit circle). - **AR.4**: Root modulus = `1.9763` (outside the unit circle). - All roots have frequencies of `-0.5000`, `-0.2472`, `0.2472`, and `0.0000`, respectively. 4. **Model Fit Metrics**: - AIC (`5184.959`), BIC (`5206.999`), and HQIC (`5193.789`) are relatively high, suggesting the model may not be optimal. - The standard deviation of innovations (`1753.258`) indicates the average error in the model’s predictions. ### Key Observations - **Non-Significant Coefficients**: Most AR terms (AR.1–AR.3) have p-values > 0.05, implying they do not significantly contribute to the model. - **Root Instability**: All roots have moduli > 1, which may indicate the model is not stable or the series is not properly differenced. - **Marginal Significance of AR.4**: The AR.4 coefficient (`0.0947`) is the only term with a p-value near 0.05, suggesting a weak but potentially relevant relationship. ### Interpretation The ARIMA(4,1,0) model for `D.Close` shows limited statistical significance in its autoregressive terms, with most coefficients failing to meet the 0.05 threshold. The roots’ moduli exceeding 1 suggest potential instability, which could undermine the model’s reliability. While the AR.4 term is marginally significant, the high AIC/BIC values indicate the model may not adequately capture the underlying data structure. The non-significant constant term further questions the model’s explanatory power. These results suggest the need for further model refinement, such as adjusting the order of differencing (d) or exploring alternative models (e.g., including moving average terms). The high standard deviation of innovations (`1753.258`) also highlights the model’s poor fit to the data.