VAR Model Overview

Multivariate Time Series Analysis

OV

VAR Model Overview

Configuration and Summary

3
N variables

VAR Model Overview and Configuration

3
n variables
2
lag order
118
n observations
21
total parameters
VAR Model
Time Series
Multivariate
IN

Key Insights

VAR Model Overview

Based on the provided VAR model overview, we are dealing with a Vector Autoregression model that includes 3 interrelated time series variables. The model is configured to use 2 lags, meaning each variable is predicted based on the previous 2 values of all variables in the system.

Key insights from the data profile:

  1. Number of Variables: The model comprises 3 variables, indicating a multivariate analysis capturing the relationships between multiple time series variables.

  2. Lag Order: The lag order is set at 2, suggesting a consideration of the past two time points of all variables to predict the current values. This accounts for dynamic dependencies between the variables over time.

  3. Number of Observations: The analysis is based on 118 observations, with each observation likely representing a distinct time point or interval.

  4. Total Parameters: The total number of parameters in the model is 21. These parameters encompass coefficients for the lagged values of the variables and intercept terms.

Insights from this model could reveal how the variables influence each other over time, potentially uncovering patterns such as lead-lag relationships or feedback dynamics among the variables. The coefficients estimated in the VAR model can provide insights into the strength and direction of these relationships.

IN

Key Insights

VAR Model Overview

Based on the provided VAR model overview, we are dealing with a Vector Autoregression model that includes 3 interrelated time series variables. The model is configured to use 2 lags, meaning each variable is predicted based on the previous 2 values of all variables in the system.

Key insights from the data profile:

  1. Number of Variables: The model comprises 3 variables, indicating a multivariate analysis capturing the relationships between multiple time series variables.

  2. Lag Order: The lag order is set at 2, suggesting a consideration of the past two time points of all variables to predict the current values. This accounts for dynamic dependencies between the variables over time.

  3. Number of Observations: The analysis is based on 118 observations, with each observation likely representing a distinct time point or interval.

  4. Total Parameters: The total number of parameters in the model is 21. These parameters encompass coefficients for the lagged values of the variables and intercept terms.

Insights from this model could reveal how the variables influence each other over time, potentially uncovering patterns such as lead-lag relationships or feedback dynamics among the variables. The coefficients estimated in the VAR model can provide insights into the strength and direction of these relationships.

SC

Stability Analysis

Model Stability Check

Model is stable
Stability status

Model Stability Analysis

Model is stable
stability status
0.95
max eigenvalue
Stable
Eigenvalues
Unit Circle
IN

Key Insights

Stability Analysis

Based on the provided data profile indicating model stability analysis, we can conclude the following insights:

  1. Model Stability: The analysis confirms that the model is stable. This is a crucial aspect for accurately predicting future outcomes over the long term.

  2. Max Eigenvalue: The maximum eigenvalue of 0.95 indicates that the model is well-behaved and likely to generate reliable forecasts. Eigenvalues are important in understanding the dynamics of a system, and a value of 0.95 suggests that the model’s behavior is well-understood and predictable.

  3. Convergence: All eigenvalues within the unit circle further assure model stability. Forecasts derived from this model are expected to converge, which is essential for ensuring that the predictions are consistent and dependable.

Overall, the model appears to be robust and suitable for making reliable long-term forecasts, given its stability characteristics and the confirmation of convergence.

IN

Key Insights

Stability Analysis

Based on the provided data profile indicating model stability analysis, we can conclude the following insights:

  1. Model Stability: The analysis confirms that the model is stable. This is a crucial aspect for accurately predicting future outcomes over the long term.

  2. Max Eigenvalue: The maximum eigenvalue of 0.95 indicates that the model is well-behaved and likely to generate reliable forecasts. Eigenvalues are important in understanding the dynamics of a system, and a value of 0.95 suggests that the model’s behavior is well-understood and predictable.

  3. Convergence: All eigenvalues within the unit circle further assure model stability. Forecasts derived from this model are expected to converge, which is essential for ensuring that the predictions are consistent and dependable.

Overall, the model appears to be robust and suitable for making reliable long-term forecasts, given its stability characteristics and the confirmation of convergence.

IC

Information Criteria

Model Selection Metrics

2
AIC/BIC

Model Selection Criteria

criterion value
AIC 7861.868
BIC 7928.365
AIC
BIC
Model Selection
IN

Key Insights

Information Criteria

Based on the data profile provided, the AIC (Akaike Information Criterion) value is 7861.87, and the BIC (Bayesian Information Criterion) value is 7928.36.

Comparing AIC and BIC values from different model specifications can help in selecting the best-fitting model that balances goodness of fit with model complexity. In this case:

  • Lower AIC and BIC values indicate better model fit after penalizing for parameter count.
  • The model with the AIC of 7861.87 and BIC of 7928.36 can be compared to other models in the selection process.
  • If there are alternative models to consider, comparing the AIC and BIC values can guide in selecting the model that best explains the data while considering the complexity of the model.

Additional analysis or model comparison could provide more insights into model selection and potentially refine the analysis further.

IN

Key Insights

Information Criteria

Based on the data profile provided, the AIC (Akaike Information Criterion) value is 7861.87, and the BIC (Bayesian Information Criterion) value is 7928.36.

Comparing AIC and BIC values from different model specifications can help in selecting the best-fitting model that balances goodness of fit with model complexity. In this case:

  • Lower AIC and BIC values indicate better model fit after penalizing for parameter count.
  • The model with the AIC of 7861.87 and BIC of 7928.36 can be compared to other models in the selection process.
  • If there are alternative models to consider, comparing the AIC and BIC values can guide in selecting the model that best explains the data while considering the complexity of the model.

Additional analysis or model comparison could provide more insights into model selection and potentially refine the analysis further.

Causal Relationships

Granger Causality and Dependencies

GC

Granger Causality Tests

Predictive Relationships

1
P-value

Granger Causality Analysis

cause effect f_statistic p_value significant
sales marketing_spend 3.450 0.040 TRUE
Causality
Predictive
Significance
IN

Key Insights

Granger Causality Tests

Based on the provided data profile, a Granger causality test was conducted, revealing a statistically significant predictive relationship in the system. Specifically, it was found that marketing spend Granger-causes sales with a p-value of 0.04.

This result suggests that changes in marketing spend can be used to predict or forecast changes in sales, indicating that marketing activities have predictive power for sales outcomes in the analyzed system. Businesses can potentially leverage this insight to optimize their marketing strategies and allocate resources effectively to drive sales growth.

If you have any more details or specific questions regarding this analysis or need further insights, feel free to provide additional information.

IN

Key Insights

Granger Causality Tests

Based on the provided data profile, a Granger causality test was conducted, revealing a statistically significant predictive relationship in the system. Specifically, it was found that marketing spend Granger-causes sales with a p-value of 0.04.

This result suggests that changes in marketing spend can be used to predict or forecast changes in sales, indicating that marketing activities have predictive power for sales outcomes in the analyzed system. Businesses can potentially leverage this insight to optimize their marketing strategies and allocate resources effectively to drive sales growth.

If you have any more details or specific questions regarding this analysis or need further insights, feel free to provide additional information.

CM

Correlation Matrix

Variable Relationships

Variable Correlation Matrix

Correlations
Heatmap
Dependencies
IN

Key Insights

Correlation Matrix

The average absolute correlation coefficient of 1 indicates perfect correlation between variables. This suggests that the variables in the dataset are highly linearly related to each other.

Given the high level of correlation, it is essential to consider employing a multivariate approach in your analysis rather than relying on separate univariate models. This approach can help capture the interdependencies and interactions between variables more accurately than individual models.

To provide more specific insights or recommendations, it would be helpful to know the variables involved in the correlation matrix or any particular goals or questions you have regarding the data.

IN

Key Insights

Correlation Matrix

The average absolute correlation coefficient of 1 indicates perfect correlation between variables. This suggests that the variables in the dataset are highly linearly related to each other.

Given the high level of correlation, it is essential to consider employing a multivariate approach in your analysis rather than relying on separate univariate models. This approach can help capture the interdependencies and interactions between variables more accurately than individual models.

To provide more specific insights or recommendations, it would be helpful to know the variables involved in the correlation matrix or any particular goals or questions you have regarding the data.

Dynamic Analysis

Impulse Response and Variance Decomposition

IR

Impulse Response Function

Shock Propagation Analysis

Impulse Response Function Analysis

IRF
Confidence Bands
Dynamic
IN

Key Insights

Impulse Response Function

Based on the data profile provided for the Impulse Response Function analysis, we can infer the following insights:

  • The peak response occurs at period 3, indicating that the impact of the shock reaches its highest value 3 periods after the initial shock.
  • The magnitude of the peak response is 1322.45, which signifies that the system’s response to the shock is substantial.
  • The positive sign of the response suggests that the shock leads to an increase in the response variable from the baseline.
  • The fact that the response dissipates over time indicates that the impact of the shock is not permanent and fades away as time progresses.
  • The data implies that the system is dynamic and responsive to external shocks, but eventually returns to its original state.

If you have any specific questions or need further analysis based on this data profile, feel free to ask!

IN

Key Insights

Impulse Response Function

Based on the data profile provided for the Impulse Response Function analysis, we can infer the following insights:

  • The peak response occurs at period 3, indicating that the impact of the shock reaches its highest value 3 periods after the initial shock.
  • The magnitude of the peak response is 1322.45, which signifies that the system’s response to the shock is substantial.
  • The positive sign of the response suggests that the shock leads to an increase in the response variable from the baseline.
  • The fact that the response dissipates over time indicates that the impact of the shock is not permanent and fades away as time progresses.
  • The data implies that the system is dynamic and responsive to external shocks, but eventually returns to its original state.

If you have any specific questions or need further analysis based on this data profile, feel free to ask!

VD

Variance Decomposition

Forecast Error Components

Forecast Error Variance Decomposition

FEVD
Contributions
Forecast
IN

Key Insights

Variance Decomposition

Thank you for providing the data profile. To further analyze the Forecast Error Variance Decomposition results, it would be beneficial to have additional details. For example:

  1. Are there specific variables of interest that significantly contribute to forecast variance?
  2. How do the contributions of different variables change over time?
  3. Are there any variables that show consistent importance across different forecast horizons?
  4. What insights or decisions are being made based on the variance decomposition results?

With this information, I can provide more tailored insights and interpretations regarding the key drivers of uncertainty in the forecast system.

IN

Key Insights

Variance Decomposition

Thank you for providing the data profile. To further analyze the Forecast Error Variance Decomposition results, it would be beneficial to have additional details. For example:

  1. Are there specific variables of interest that significantly contribute to forecast variance?
  2. How do the contributions of different variables change over time?
  3. Are there any variables that show consistent importance across different forecast horizons?
  4. What insights or decisions are being made based on the variance decomposition results?

With this information, I can provide more tailored insights and interpretations regarding the key drivers of uncertainty in the forecast system.

Forecasting Performance

Historical Fit and Future Predictions

HF

Historical Fit

Actual vs Fitted Values

Historical Fit Analysis

In-sample
Fit Quality
Comparison
IN

Key Insights

Historical Fit

Thank you for providing the data profile for historical fit analysis. From the summary, it seems that the analysis involves comparing model predictions to actual values across 118 time periods. Here are some insights and recommendations based on the data profile:

  1. Performance Evaluation:

    • Evaluate the performance of the model by calculating measures such as Mean Absolute Error (MAE), Mean Squared Error (MSE), Root Mean Squared Error (RMSE), or R-squared to quantify the accuracy of the predictions compared to actual values.

  2. Residual Analysis:

    • Conduct a thorough analysis of residuals by plotting them against the fitted values to check for randomness.
    • Look for any systematic patterns or trends in the residuals that could indicate issues with the model assumptions or data quality.

  3. Outlier Detection:

    • Identify and investigate any large residuals that may suggest the presence of outliers or structural breaks in the data.
    • Outliers could impact the model’s performance and may require special treatment or further investigation.

  4. Model Improvement:

    • If the residuals exhibit patterns or if the model performance is not satisfactory, consider refining the model by incorporating additional features, adjusting parameters, or trying a different modeling approach.

  5. Visual Diagnostic Tools:

    • Use diagnostic plots such as residual plots, Q-Q plots, or time series plots to visually assess the goodness of fit and identify any potential issues in the model’s performance.

  6. Time Series Analysis:

    • Given the time series nature of the data, consider time series analysis techniques like autocorrelation analysis or seasonal decomposition to understand the underlying patterns and improve the model’s predictive power.

If there are any specific details or results from the analysis that you would like to delve into further, please provide additional information for a more detailed interpretation.

IN

Key Insights

Historical Fit

Thank you for providing the data profile for historical fit analysis. From the summary, it seems that the analysis involves comparing model predictions to actual values across 118 time periods. Here are some insights and recommendations based on the data profile:

  1. Performance Evaluation:

    • Evaluate the performance of the model by calculating measures such as Mean Absolute Error (MAE), Mean Squared Error (MSE), Root Mean Squared Error (RMSE), or R-squared to quantify the accuracy of the predictions compared to actual values.

  2. Residual Analysis:

    • Conduct a thorough analysis of residuals by plotting them against the fitted values to check for randomness.
    • Look for any systematic patterns or trends in the residuals that could indicate issues with the model assumptions or data quality.

  3. Outlier Detection:

    • Identify and investigate any large residuals that may suggest the presence of outliers or structural breaks in the data.
    • Outliers could impact the model’s performance and may require special treatment or further investigation.

  4. Model Improvement:

    • If the residuals exhibit patterns or if the model performance is not satisfactory, consider refining the model by incorporating additional features, adjusting parameters, or trying a different modeling approach.

  5. Visual Diagnostic Tools:

    • Use diagnostic plots such as residual plots, Q-Q plots, or time series plots to visually assess the goodness of fit and identify any potential issues in the model’s performance.

  6. Time Series Analysis:

    • Given the time series nature of the data, consider time series analysis techniques like autocorrelation analysis or seasonal decomposition to understand the underlying patterns and improve the model’s predictive power.

If there are any specific details or results from the analysis that you would like to delve into further, please provide additional information for a more detailed interpretation.

FC

Multi-step Forecasts

Future Predictions

Multi-step Ahead Forecasts

Forecasts
Confidence
Multi-step
IN

Key Insights

Multi-step Forecasts

Thank you for providing the data profile. The 12-period ahead forecasts for 3 variables, considering interdependencies between the variables, can provide valuable insights into future trends and potential outcomes. Here are some key points and insights based on the data profile:

  1. Long-Term Forecasting: The model’s capability to generate forecasts for 12 periods ahead allows for long-term planning and decision-making. This extended horizon can be particularly useful for strategic planning and identifying trends that may not be apparent in short-term forecasts.

  2. Interdependencies Between Variables: By accounting for the interdependencies between the variables, the model can capture the complex relationships and interactions that exist within the dataset. Understanding how changes in one variable may impact the others can lead to more accurate and reliable forecasts.

  3. Increasing Uncertainty: The widening confidence intervals at longer horizons indicate a growing level of uncertainty as the forecasting horizon extends. This is a common attribute in forecasting, where predictions become less precise as the time horizon increases due to various external factors and unforeseen events.

  4. Risk Management: The recognition of increasing uncertainty in longer-term forecasts highlights the importance of risk management and scenario planning. Decision-makers can use this information to assess potential risks and develop strategies to mitigate them.

  5. Monitoring and Evaluation: Continuous monitoring and evaluation of the forecasts against actual outcomes can help in refining the model and improving its accuracy over time. Regularly updating the forecasts based on new data and feedback can enhance the model’s reliability.

If you have specific questions or require further analysis based on the detailed forecast data, feel free to provide additional information for a more in-depth examination.

IN

Key Insights

Multi-step Forecasts

Thank you for providing the data profile. The 12-period ahead forecasts for 3 variables, considering interdependencies between the variables, can provide valuable insights into future trends and potential outcomes. Here are some key points and insights based on the data profile:

  1. Long-Term Forecasting: The model’s capability to generate forecasts for 12 periods ahead allows for long-term planning and decision-making. This extended horizon can be particularly useful for strategic planning and identifying trends that may not be apparent in short-term forecasts.

  2. Interdependencies Between Variables: By accounting for the interdependencies between the variables, the model can capture the complex relationships and interactions that exist within the dataset. Understanding how changes in one variable may impact the others can lead to more accurate and reliable forecasts.

  3. Increasing Uncertainty: The widening confidence intervals at longer horizons indicate a growing level of uncertainty as the forecasting horizon extends. This is a common attribute in forecasting, where predictions become less precise as the time horizon increases due to various external factors and unforeseen events.

  4. Risk Management: The recognition of increasing uncertainty in longer-term forecasts highlights the importance of risk management and scenario planning. Decision-makers can use this information to assess potential risks and develop strategies to mitigate them.

  5. Monitoring and Evaluation: Continuous monitoring and evaluation of the forecasts against actual outcomes can help in refining the model and improving its accuracy over time. Regularly updating the forecasts based on new data and feedback can enhance the model’s reliability.

If you have specific questions or require further analysis based on the detailed forecast data, feel free to provide additional information for a more in-depth examination.

Model Diagnostics

Residual Analysis and Model Fit

MF

Model Fit Statistics

R-squared by Equation

3
R²

Model Fit Statistics by Equation

equation r_squared adj_r_squared
sales 0.750 0.730
marketing_spend 0.750 0.730
customer_acquisition 0.750 0.730
Fit Quality
R-squared
F-statistics
IN

Key Insights

Model Fit Statistics

The average R-squared value of 0.75 across all equations indicates a strong level of explanatory power in the models. This means that, on average, the models capture approximately 75% of the variation in the data.

However, to provide more specific insights and recommendations, it would be helpful to have additional information such as the individual R-squared values for each equation, the corresponding F-statistics for each equation to test overall model significance, and any specific questions or goals related to the analysis of these models.

IN

Key Insights

Model Fit Statistics

The average R-squared value of 0.75 across all equations indicates a strong level of explanatory power in the models. This means that, on average, the models capture approximately 75% of the variation in the data.

However, to provide more specific insights and recommendations, it would be helpful to have additional information such as the individual R-squared values for each equation, the corresponding F-statistics for each equation to test overall model significance, and any specific questions or goals related to the analysis of these models.

RD

Residual Diagnostics

Autocorrelation Analysis

Residual Diagnostics and Tests

ACF
Residuals
Independence
IN

Key Insights

Residual Diagnostics

Based on the provided data profile, it seems that the residual diagnostics of the model have been conducted, focusing on assessing model adequacy with respect to serial correlation. The analysis indicates that no serial correlation has been detected, suggesting that the model adequately captures the temporal dynamics in the data.

Furthermore, the use of ACF (Autocorrelation Function) and PACF (Partial Autocorrelation Function) plots is highlighted as a technique to identify any remaining autocorrelation patterns that may exist in the residuals. These plots can help in fine-tuning the model and ensuring that it adequately captures the relationships present in the data.

Overall, the absence of serial correlation and the emphasis on using ACF and PACF plots suggest a robust approach to residual diagnostics and model validation. It indicates that the model is likely capturing the key dynamics of the data effectively.

IN

Key Insights

Residual Diagnostics

Based on the provided data profile, it seems that the residual diagnostics of the model have been conducted, focusing on assessing model adequacy with respect to serial correlation. The analysis indicates that no serial correlation has been detected, suggesting that the model adequately captures the temporal dynamics in the data.

Furthermore, the use of ACF (Autocorrelation Function) and PACF (Partial Autocorrelation Function) plots is highlighted as a technique to identify any remaining autocorrelation patterns that may exist in the residuals. These plots can help in fine-tuning the model and ensuring that it adequately captures the relationships present in the data.

Overall, the absence of serial correlation and the emphasis on using ACF and PACF plots suggest a robust approach to residual diagnostics and model validation. It indicates that the model is likely capturing the key dynamics of the data effectively.

LS

Lag Selection

Optimal Lag Order

1
Lag

Optimal Lag Selection

criterion optimal_lag
Default 2.000
Lag Order
Optimal
Selection
IN

Key Insights

Lag Selection

Based on the provided data profile, the optimal lag order selected is 2 based on the default method specified by the user. This selection strikes a balance between capturing important temporal dependencies in the data while maintaining model simplicity and forecast accuracy. With a lag order of 2, the model is expected to capture essential dynamic relationships without introducing unnecessary complexity or overfitting. This decision is crucial for ensuring that the model can effectively capture the underlying patterns and dynamics in the data for accurate forecasting or analysis.

IN

Key Insights

Lag Selection

Based on the provided data profile, the optimal lag order selected is 2 based on the default method specified by the user. This selection strikes a balance between capturing important temporal dependencies in the data while maintaining model simplicity and forecast accuracy. With a lag order of 2, the model is expected to capture essential dynamic relationships without introducing unnecessary complexity or overfitting. This decision is crucial for ensuring that the model can effectively capture the underlying patterns and dynamics in the data for accurate forecasting or analysis.

Stability and Selection

Model Stability and Information Criteria

IR

Impulse Response Function

Shock Propagation Analysis

Impulse Response Function Analysis

IRF
Confidence Bands
Dynamic
IN

Key Insights

Impulse Response Function

Based on the data profile provided for the Impulse Response Function analysis, we can infer the following insights:

  • The peak response occurs at period 3, indicating that the impact of the shock reaches its highest value 3 periods after the initial shock.
  • The magnitude of the peak response is 1322.45, which signifies that the system’s response to the shock is substantial.
  • The positive sign of the response suggests that the shock leads to an increase in the response variable from the baseline.
  • The fact that the response dissipates over time indicates that the impact of the shock is not permanent and fades away as time progresses.
  • The data implies that the system is dynamic and responsive to external shocks, but eventually returns to its original state.

If you have any specific questions or need further analysis based on this data profile, feel free to ask!

IN

Key Insights

Impulse Response Function

Based on the data profile provided for the Impulse Response Function analysis, we can infer the following insights:

  • The peak response occurs at period 3, indicating that the impact of the shock reaches its highest value 3 periods after the initial shock.
  • The magnitude of the peak response is 1322.45, which signifies that the system’s response to the shock is substantial.
  • The positive sign of the response suggests that the shock leads to an increase in the response variable from the baseline.
  • The fact that the response dissipates over time indicates that the impact of the shock is not permanent and fades away as time progresses.
  • The data implies that the system is dynamic and responsive to external shocks, but eventually returns to its original state.

If you have any specific questions or need further analysis based on this data profile, feel free to ask!

Technical Details

Coefficients and Model Specifications

CS

Model Coefficients

Detailed Estimates

9
Coefficients

Model Coefficients Summary

equation variable estimate std_error t_value p_value
sales const 0.377 0.089 -0.055 0.144
sales lag1 0.357 0.090 -0.187 0.481
sales lag2 -0.630 0.044 0.611 0.822
marketing_spend const -0.440 0.017 0.344 0.770
marketing_spend lag1 -0.952 0.026 -0.020 0.137
marketing_spend lag2 -0.421 0.024 -0.444 0.996
customer_acquisition const -0.049 0.158 -2.670 0.825
customer_acquisition lag1 0.774 0.007 -1.355 0.505
customer_acquisition lag2 1.683 0.018 -0.754 0.655
Coefficients
P-values
Significance
IN

Key Insights

Model Coefficients

From the provided data profile, we see that there are a total of 9 coefficients estimated across all equations. Interestingly, none of the coefficients are significant at the 5% level, indicating that there are no strong dependencies between the lagged variables in the model.

The lack of significant coefficients suggests that the past values of the variables may not have a strong influence on the current values in the model. This finding could imply that the variables being considered may not exhibit a clear pattern of influence from their lagged values.

Further analysis or exploration may be needed to understand the relationships between the variables better, as the current coefficients do not show significant dependencies. Additional context or information about the variables and the model could help in interpreting these results more comprehensively.

IN

Key Insights

Model Coefficients

From the provided data profile, we see that there are a total of 9 coefficients estimated across all equations. Interestingly, none of the coefficients are significant at the 5% level, indicating that there are no strong dependencies between the lagged variables in the model.

The lack of significant coefficients suggests that the past values of the variables may not have a strong influence on the current values in the model. This finding could imply that the variables being considered may not exhibit a clear pattern of influence from their lagged values.

Further analysis or exploration may be needed to understand the relationships between the variables better, as the current coefficients do not show significant dependencies. Additional context or information about the variables and the model could help in interpreting these results more comprehensively.

Job ID: test_analytics__time_series__multivariate__var_20250915_194808
Secret Hash: 36f2d40ab80e