December 8,2023
The ARIMA (Autoregressive Integrated Moving Average) model is a popular statistical method for time series forecasting that captures the dynamics of the series through three main parameters: AR (p), I (d), and MA (q).
AR (Autoregression): Refers to the use of previous values in the time series to predict future values.
I (Integrated): Represents the differencing of raw observations to make the time series stationary, which means that the series has constant mean and variance over time.
MA (Moving Average): Incorporates the dependency between an observation and a residual error from a moving average model applied to lagged observations.
The lines in each graph represent:
Blue Line (Actual): The actual observed values from the dataset.
Orange Line (Fitted): The values predicted by the ARIMA model.
Logan Passengers: The ARIMA model appears to track the seasonal pattern and general trend of the passenger data quite closely.
Logan International Flights: The model captures the seasonality and fluctuations in the number of flights well.
Hotel Occupancy Rate: The ARIMA model follows the actual occupancy rates, including the seasonal peaks and troughs.
Hotel Average Daily Rate: The model fits the average daily rate with some accuracy, again reflecting the seasonality in the data.
Total Jobs: The ARIMA model fits the data tightly, suggesting that the model is capturing the underlying trend effectively.
Unemployment Rate: The model seems to follow the actual rate closely, including the downward trend over time.
Labor Force Participation Rate: The ARIMA model provides a reasonable fit to the data, capturing the stability of the participation rate over time.
December 6,2023
The Partial Autocorrelation Function (PACF) plots show the partial correlation of each time series with its own lagged values, controlling for the values of the time series at all shorter lags. This is helpful in identifying the order of the autoregressive (AR) part of an ARIMA model. Here’s how to interpret these plots:
Logan Passengers: The PACF plot for ‘Logan Passengers’ might show significant partial autocorrelations at one or more lags. Significant spikes (those that cross the blue confidence interval) suggest that those lags have a predictive relationship with the current value, after accounting for the relationships at all shorter lags. If such spikes occur at the first few lags and then cut off, it indicates an AR process of that order.
Logan Intl Flights: Like ‘Logan Passengers’, look for significant spikes in the early lags. The number of significant lags can indicate the order of an AR process for ‘Logan Intl Flights’. If there are no significant spikes or they are sporadic, it might suggest that an AR process is not appropriate.
Hotel Occupancy Rate: If there are significant spikes at fixed intervals, it may suggest seasonality in the data. Otherwise, the number and position of significant spikes can help determine the order of the AR process.
Labor Force Participation Rate: This PACF plot would be analyzed in the same manner, identifying the number of significant lags to determine the potential order of an AR process.
Hotel Average Daily Rate: If significant partial autocorrelations are present, they indicate the potential order of the AR process. If they decay gradually, it might suggest a mixed ARMA process.
Total Jobs: Look for the point at which the partial autocorrelations become insignificant. This will give you the suggested order of the AR process for the ‘Total Jobs’ series.
Unemployment Rate: As with the others, the presence and position of significant partial autocorrelations will inform the choice of AR order for modeling the ‘Unemployment Rate’.
December 4,2023
Autocorrelation Function (ACF) plots for various time series data. These plots show the correlation of the time series with its own past values at different lags (time intervals). The blue shaded area in the plot represents the confidence interval, typically set at 95%. Correlation values outside of this area are considered statistically significant.
Here’s a brief explanation for each graph:
Logan Passengers: The ACF plot for ‘Logan Passengers’ shows that there are a few lags where the correlation is significant. This suggests that past values of the series have some correlation with future values, hinting at a potential AR process.
Logan Intl Flights: The ‘Logan Intl Flights’ ACF plot indicates significant autocorrelation at a few initial lags. This might suggest an autoregressive component in the time series, which could be used in model identification.
Hotel Occupancy Rate: This plot shows several significant spikes, suggesting a strong seasonal pattern or an AR component that repeats at regular intervals.
Labor Force Participation Rate: The ACF for ‘Labor Force Participation Rate’ shows fewer significant correlations, indicating that the series might not be strongly dependent on its past values, or it might be a more complex model that does not fit neatly into an AR or MA process.
Hotel Average Daily Rate: The plot displays very few significant correlations at specific lags, which may suggest that the series has a less pronounced AR structure.
Total Jobs: Significant correlations between a few initial lags are visible, which could indicate an AR process at work. The data might be influenced by its values in the near past.
Unemployment Rate: The ACF plot shows almost no significant autocorrelation at any lag, suggesting that past values do not have a strong linear relationship with future values.
December 1, 2023
Time series plots of differenced economic indicators, each with a p-value indicating the significance of a statistical test, likely a unit root test such as the Augmented Dickey-Fuller (ADF) test. Here’s an analysis of what each plot represents:
Differenced logan_passengers: This plot shows the changes in the number of passengers over time after differencing the data (likely to achieve stationarity). The p-value suggests that the differenced series is stationary (p < 0.05).
Differenced logan_intl_flights: Like the passengers’ plot, this shows the changes in the number of international flights. The p-value is above the common threshold of 0.05, suggesting that the series may not be stationary.
Differenced hotel_occup_rate: This graph displays the changes in the hotel occupancy rate over time. The p- value is 0.0000, which is highly significant and indicates stationarity of the differenced series.
Differenced hotel_avg_daily_rate: Shows the changes in the average daily rate for hotels. The p-value again indicates that the differenced series is stationary.
Differenced total_jobs: This represents the changes in the total number of jobs. The p-value is not below the 0.05 threshold, suggesting non-stationarity.
Differenced unemp_rate: The changes in the unemployment rate over time are plotted here. The p-value is greater than 0.05, suggesting that the series may not be stationary.
Differenced labor_force_part_rate: Shows the changes in the labor force participation rate. The p-value is close to the threshold, which could suggest marginal stationarity depending on the specific significance level you are using.
In each plot, the time series data have been different, which is a common technique to remove trends and seasonal patterns to achieve stationarity in time series analysis. Stationarity is an important assumption in many time series models, and the ADF test is often used to test for it. The low p-value (typically <0.05) in the ADF test suggests that the null hypothesis of the presence of a unit root can be rejected, implying stationarity.