document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1167135>Question 1167135</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Data.xls contains quarterly data on sales of refrigerators and durable goods for the period 1978 to 1995.<BR>\n" );
document.write( "a.Estimate the following dummy variable models and interpret the results. FRIG= β1+β2D2+ β3D3 + β4D4+ μDUR= β1+β2D2+ β3D3 + β4D4+ μ<BR>\n" );
document.write( "b.Introduce a dummy variable D1 representing sales in the first quarter. Then, re-estimate the following models and interpret the results. FRIG = β1D1 +β2D2 + β3D3 + β4D4 + μDUR = β1D1 +β2D2+ β3D3 + β4D4 + μ<BR>\n" );
document.write( "c.De-seasonalize the sales of refrigerators and durable goods using the three step seasonal adjustment procedure. Explain the procedureand purpose of de-seasonalization.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "FRIG	DUR	D2	D3	D4<BR>\n" );
document.write( "1317	252.6	0	0	0<BR>\n" );
document.write( "1615	272.4	1	0	0<BR>\n" );
document.write( "1662	270.9	0	1	0<BR>\n" );
document.write( "1295	273.9	0	0	1<BR>\n" );
document.write( "1271	268.9	0	0	0<BR>\n" );
document.write( "1555	262.9	1	0	0<BR>\n" );
document.write( "1639	270.9	0	1	0<BR>\n" );
document.write( "1238	263.4	0	0	1<BR>\n" );
document.write( "1277	260.6	0	0	0<BR>\n" );
document.write( "1258	231.9	1	0	0<BR>\n" );
document.write( "1417	242.7	0	1	0<BR>\n" );
document.write( "1185	248.6	0	0	1<BR>\n" );
document.write( "1196	258.7	0	0	0<BR>\n" );
document.write( "1410	248.4	1	0	0<BR>\n" );
document.write( "1417	255.5	0	1	0<BR>\n" );
document.write( "919	240.4	0	0	1<BR>\n" );
document.write( "943	247.7	0	0	0<BR>\n" );
document.write( "1175	249.1	1	0	0<BR>\n" );
document.write( "1269	251.8	0	1	0<BR>\n" );
document.write( "973	262	0	0	1<BR>\n" );
document.write( "1102	263.3	0	0	0<BR>\n" );
document.write( "1344	280	1	0	0<BR>\n" );
document.write( "1641	288.5	0	1	0<BR>\n" );
document.write( "1225	300.5	0	0	1<BR>\n" );
document.write( "1429	312.6	0	0	0<BR>\n" );
document.write( "1699	322.5	1	0	0<BR>\n" );
document.write( "1749	324.3	0	1	0<BR>\n" );
document.write( "1117	333.1	0	0	1<BR>\n" );
document.write( "1242	344.8	0	0	0<BR>\n" );
document.write( "1684	350.3	1	0	0<BR>\n" );
document.write( "1764	369.1	0	1	0<BR>\n" );
document.write( "1328	356.4	0	0	1<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852339 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852339\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import pandas as pd<BR>\n" );
document.write( "import numpy as np<BR>\n" );
document.write( "import statsmodels.api as sm\r<BR>\n" );
document.write( "\n" );
document.write( "# Provided data<BR>\n" );
document.write( "data_str = \"\"\"<BR>\n" );
document.write( "FRIG DUR D2 D3 D4<BR>\n" );
document.write( "1317 252.6 0 0 0<BR>\n" );
document.write( "1615 272.4 1 0 0<BR>\n" );
document.write( "1662 270.9 0 1 0<BR>\n" );
document.write( "1295 273.9 0 0 1<BR>\n" );
document.write( "1271 268.9 0 0 0<BR>\n" );
document.write( "1555 262.9 1 0 0<BR>\n" );
document.write( "1639 270.9 0 1 0<BR>\n" );
document.write( "1238 263.4 0 0 1<BR>\n" );
document.write( "1277 260.6 0 0 0<BR>\n" );
document.write( "1258 231.9 1 0 0<BR>\n" );
document.write( "1417 242.7 0 1 0<BR>\n" );
document.write( "1185 248.6 0 0 1<BR>\n" );
document.write( "1196 258.7 0 0 0<BR>\n" );
document.write( "1410 248.4 1 0 0<BR>\n" );
document.write( "1417 255.5 0 1 0<BR>\n" );
document.write( "919 240.4 0 0 1<BR>\n" );
document.write( "943 247.7 0 0 0<BR>\n" );
document.write( "1175 249.1 1 0 0<BR>\n" );
document.write( "1269 251.8 0 1 0<BR>\n" );
document.write( "973 262.0 0 0 1<BR>\n" );
document.write( "1102 263.3 0 0 0<BR>\n" );
document.write( "1344 280.0 1 0 0<BR>\n" );
document.write( "1641 288.5 0 1 0<BR>\n" );
document.write( "1225 300.5 0 0 1<BR>\n" );
document.write( "1429 312.6 0 0 0<BR>\n" );
document.write( "1699 322.5 1 0 0<BR>\n" );
document.write( "1749 324.3 0 1 0<BR>\n" );
document.write( "1117 333.1 0 0 1<BR>\n" );
document.write( "1242 344.8 0 0 0<BR>\n" );
document.write( "1684 350.3 1 0 0<BR>\n" );
document.write( "1764 369.1 0 1 0<BR>\n" );
document.write( "1328 356.4 0 0 1<BR>\n" );
document.write( "\"\"\"\r<BR>\n" );
document.write( "\n" );
document.write( "# Convert data to DataFrame<BR>\n" );
document.write( "from io import StringIO<BR>\n" );
document.write( "df = pd.read_csv(StringIO(data_str), sep='\s+')\r<BR>\n" );
document.write( "\n" );
document.write( "# Create D1 for part b and c<BR>\n" );
document.write( "df['D1'] = 1 - df['D2'] - df['D3'] - df['D4']\r<BR>\n" );
document.write( "\n" );
document.write( "# --- A) Estimate models with INTERCEPT (D1 is reference) ---<BR>\n" );
document.write( "X_a = sm.add_constant(df[['D2', 'D3', 'D4']], prepend=True)\r<BR>\n" );
document.write( "\n" );
document.write( "# FRIG model (a)<BR>\n" );
document.write( "model_frig_a = sm.OLS(df['FRIG'], X_a).fit()<BR>\n" );
document.write( "# DUR model (a)<BR>\n" );
document.write( "model_dur_a = sm.OLS(df['DUR'], X_a).fit()\r<BR>\n" );
document.write( "\n" );
document.write( "# --- B) Estimate models without INTERCEPT (D1, D2, D3, D4 included) ---<BR>\n" );
document.write( "X_b = df[['D1', 'D2', 'D3', 'D4']]\r<BR>\n" );
document.write( "\n" );
document.write( "# FRIG model (b)<BR>\n" );
document.write( "model_frig_b = sm.OLS(df['FRIG'], X_b).fit()<BR>\n" );
document.write( "# DUR model (b)<BR>\n" );
document.write( "model_dur_b = sm.OLS(df['DUR'], X_b).fit()\r<BR>\n" );
document.write( "\n" );
document.write( "# --- C) De-seasonalization (Using coefficients from model_b) ---\r<BR>\n" );
document.write( "\n" );
document.write( "# 1. Estimate Seasonal Components (Quarterly Means)<BR>\n" );
document.write( "# FRIG Seasonal Components (Si_frig)<BR>\n" );
document.write( "Si_frig = model_frig_b.params.to_dict()<BR>\n" );
document.write( "# DUR Seasonal Components (Si_dur)<BR>\n" );
document.write( "Si_dur = model_dur_b.params.to_dict()\r<BR>\n" );
document.write( "\n" );
document.write( "# 2. Calculate Overall Mean<BR>\n" );
document.write( "mean_frig = df['FRIG'].mean()<BR>\n" );
document.write( "mean_dur = df['DUR'].mean()\r<BR>\n" );
document.write( "\n" );
document.write( "# 3. Calculate Seasonal Index (SI_i = S_i - Overall Mean)<BR>\n" );
document.write( "SI_frig = {k: v - mean_frig for k, v in Si_frig.items()}<BR>\n" );
document.write( "SI_dur = {k: v - mean_dur for k, v in Si_dur.items()}\r<BR>\n" );
document.write( "\n" );
document.write( "# Map seasonal indices back to the DataFrame rows<BR>\n" );
document.write( "df['SI_FRIG'] = (df['D1'] * SI_frig['D1'] + df['D2'] * SI_frig['D2'] +<BR>\n" );
document.write( "                 df['D3'] * SI_frig['D3'] + df['D4'] * SI_frig['D4'])<BR>\n" );
document.write( "df['SI_DUR'] = (df['D1'] * SI_dur['D1'] + df['D2'] * SI_dur['D2'] +<BR>\n" );
document.write( "                df['D3'] * SI_dur['D3'] + df['D4'] * SI_dur['D4'])\r<BR>\n" );
document.write( "\n" );
document.write( "# 4. De-seasonalize the sales (De-seasonalized = Original - Seasonal Index)<BR>\n" );
document.write( "df['FRIG_DESEASONALIZED'] = df['FRIG'] - df['SI_FRIG']<BR>\n" );
document.write( "df['DUR_DESEASONALIZED'] = df['DUR'] - df['SI_DUR']\r<BR>\n" );
document.write( "\n" );
document.write( "# Prepare results for the user<BR>\n" );
document.write( "results = {<BR>\n" );
document.write( "    'model_frig_a': model_frig_a.summary().as_text(),<BR>\n" );
document.write( "    'model_dur_a': model_dur_a.summary().as_text(),<BR>\n" );
document.write( "    'model_frig_b': model_frig_b.summary().as_text(),<BR>\n" );
document.write( "    'model_dur_b': model_dur_b.summary().as_text(),<BR>\n" );
document.write( "    'mean_frig': mean_frig,<BR>\n" );
document.write( "    'mean_dur': mean_dur,<BR>\n" );
document.write( "    'SI_frig': SI_frig,<BR>\n" );
document.write( "    'SI_dur': SI_dur,<BR>\n" );
document.write( "    'deseasonalized_data': df[['FRIG', 'FRIG_DESEASONALIZED', 'DUR', 'DUR_DESEASONALIZED']].to_csv('deseasonalized_data.csv', index=False)<BR>\n" );
document.write( "}\r<BR>\n" );
document.write( "\n" );
document.write( "print(f\"FRIG Model (a) Summary:\n{results['model_frig_a']}\")<BR>\n" );
document.write( "print(f\"\nDUR Model (a) Summary:\n{results['model_dur_a']}\")<BR>\n" );
document.write( "print(f\"\nFRIG Model (b) Summary:\n{results['model_frig_b']}\")<BR>\n" );
document.write( "print(f\"\nDUR Model (b) Summary:\n{results['model_dur_b']}\")<BR>\n" );
document.write( "print(f\"\nFRIG Seasonal Indices (SI): {results['SI_frig']}\")<BR>\n" );
document.write( "print(f\"DUR Seasonal Indices (SI): {results['SI_dur']}\")<BR>\n" );
document.write( "print(\"De-seasonalized data saved to deseasonalized_data.csv\")\r<BR>\n" );
document.write( "\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "FRIG Model (a) Summary:<BR>\n" );
document.write( "                            OLS Regression Results                            <BR>\n" );
document.write( "==============================================================================<BR>\n" );
document.write( "Dep. Variable:                   FRIG   R-squared:                       0.532<BR>\n" );
document.write( "Model:                            OLS   Adj. R-squared:                  0.482<BR>\n" );
document.write( "Method:                 Least Squares   F-statistic:                     10.60<BR>\n" );
document.write( "Date:                Sun, 19 Oct 2025   Prob (F-statistic):           7.91e-05<BR>\n" );
document.write( "Time:                        20:32:45   Log-Likelihood:                -207.55<BR>\n" );
document.write( "No. Observations:                  32   AIC:                             423.1<BR>\n" );
document.write( "Df Residuals:                      28   BIC:                             429.0<BR>\n" );
document.write( "Df Model:                           3                                         <BR>\n" );
document.write( "Covariance Type:            nonrobust                                         <BR>\n" );
document.write( "==============================================================================<BR>\n" );
document.write( "                 coef    std err          t      P>|t|      [0.025      0.975]<BR>\n" );
document.write( "------------------------------------------------------------------------------<BR>\n" );
document.write( "const       1222.1250     59.990     20.372      0.000    1099.240    1345.010<BR>\n" );
document.write( "D2           245.3750     84.839      2.892      0.007      71.590     419.160<BR>\n" );
document.write( "D3           347.6250     84.839      4.097      0.000     173.840     521.410<BR>\n" );
document.write( "D4           -62.1250     84.839     -0.732      0.470    -235.910     111.660<BR>\n" );
document.write( "==============================================================================<BR>\n" );
document.write( "Omnibus:                        3.849   Durbin-Watson:                   0.393<BR>\n" );
document.write( "Prob(Omnibus):                  0.146   Jarque-Bera (JB):                2.306<BR>\n" );
document.write( "Skew:                          -0.436   Prob(JB):                        0.316<BR>\n" );
document.write( "Kurtosis:                       2.015   Cond. No.                         4.79<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Notes:<BR>\n" );
document.write( "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\r<BR>\n" );
document.write( "\n" );
document.write( "DUR Model (a) Summary:<BR>\n" );
document.write( "                            OLS Regression Results                            <BR>\n" );
document.write( "==============================================================================<BR>\n" );
document.write( "Dep. Variable:                    DUR   R-squared:                       0.011<BR>\n" );
document.write( "Model:                            OLS   Adj. R-squared:                 -0.095<BR>\n" );
document.write( "Method:                 Least Squares   F-statistic:                    0.1045<BR>\n" );
document.write( "Date:                Sun, 19 Oct 2025   Prob (F-statistic):              0.957<BR>\n" );
document.write( "Time:                        20:32:45   Log-Likelihood:                -161.15<BR>\n" );
document.write( "No. Observations:                  32   AIC:                             330.3<BR>\n" );
document.write( "Df Residuals:                      28   BIC:                             336.2<BR>\n" );
document.write( "Df Model:                           3                                         <BR>\n" );
document.write( "Covariance Type:            nonrobust                                         <BR>\n" );
document.write( "==============================================================================<BR>\n" );
document.write( "                 coef    std err          t      P>|t|      [0.025      0.975]<BR>\n" );
document.write( "------------------------------------------------------------------------------<BR>\n" );
document.write( "const        276.1500     14.069     19.628      0.000     247.330     304.970<BR>\n" );
document.write( "D2             1.0375     19.897      0.052      0.959     -39.720      41.795<BR>\n" );
document.write( "D3             8.0625     19.897      0.405      0.688     -32.695      48.820<BR>\n" );
document.write( "D4             8.6375     19.897      0.434      0.668     -32.120      49.395<BR>\n" );
document.write( "==============================================================================<BR>\n" );
document.write( "Omnibus:                        5.154   Durbin-Watson:                   0.100<BR>\n" );
document.write( "Prob(Omnibus):                  0.076   Jarque-Bera (JB):                4.808<BR>\n" );
document.write( "Skew:                           0.930   Prob(JB):                       0.0904<BR>\n" );
document.write( "Kurtosis:                       2.615   Cond. No.                         4.79<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Notes:<BR>\n" );
document.write( "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\r<BR>\n" );
document.write( "\n" );
document.write( "FRIG Model (b) Summary:<BR>\n" );
document.write( "                            OLS Regression Results                            <BR>\n" );
document.write( "==============================================================================<BR>\n" );
document.write( "Dep. Variable:                   FRIG   R-squared:                       0.532<BR>\n" );
document.write( "Model:                            OLS   Adj. R-squared:                  0.482<BR>\n" );
document.write( "Method:                 Least Squares   F-statistic:                     10.60<BR>\n" );
document.write( "Date:                Sun, 19 Oct 2025   Prob (F-statistic):           7.91e-05<BR>\n" );
document.write( "Time:                        20:32:45   Log-Likelihood:                -207.55<BR>\n" );
document.write( "No. Observations:                  32   AIC:                             423.1<BR>\n" );
document.write( "Df Residuals:                      28   BIC:                             429.0<BR>\n" );
document.write( "Df Model:                           3                                         <BR>\n" );
document.write( "Covariance Type:            nonrobust                                         <BR>\n" );
document.write( "==============================================================================<BR>\n" );
document.write( "                 coef    std err          t      P>|t|      [0.025      0.975]<BR>\n" );
document.write( "------------------------------------------------------------------------------<BR>\n" );
document.write( "D1          1222.1250     59.990     20.372      0.000    1099.240    1345.010<BR>\n" );
document.write( "D2          1467.5000     59.990     24.462      0.000    1344.615    1590.385<BR>\n" );
document.write( "D3          1569.7500     59.990     26.167      0.000    1446.865    1692.635<BR>\n" );
document.write( "D4          1160.0000     59.990     19.336      0.000    1037.115    1282.885<BR>\n" );
document.write( "==============================================================================<BR>\n" );
document.write( "Omnibus:                        3.849   Durbin-Watson:                   0.393<BR>\n" );
document.write( "Prob(Omnibus):                  0.146   Jarque-Bera (JB):                2.306<BR>\n" );
document.write( "Skew:                          -0.436   Prob(JB):                        0.316<BR>\n" );
document.write( "Kurtosis:                       2.015   Cond. No.                         1.00<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Notes:<BR>\n" );
document.write( "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\r<BR>\n" );
document.write( "\n" );
document.write( "DUR Model (b) Summary:<BR>\n" );
document.write( "                            OLS Regression Results                            <BR>\n" );
document.write( "==============================================================================<BR>\n" );
document.write( "Dep. Variable:                    DUR   R-squared:                       0.011<BR>\n" );
document.write( "Model:                            OLS   Adj. R-squared:                 -0.095<BR>\n" );
document.write( "Method:                 Least Squares   F-statistic:                    0.1045<BR>\n" );
document.write( "Date:                Sun, 19 Oct 2025   Prob (F-statistic):              0.957<BR>\n" );
document.write( "Time:                        20:32:45   Log-Likelihood:                -161.15<BR>\n" );
document.write( "No. Observations:                  32   AIC:                             330.3<BR>\n" );
document.write( "Df Residuals:                      28   BIC:                             336.2<BR>\n" );
document.write( "Df Model:                           3                                         <BR>\n" );
document.write( "Covariance Type:            nonrobust                                         <BR>\n" );
document.write( "==============================================================================<BR>\n" );
document.write( "                 coef    std err          t      P>|t|      [0.025      0.975]<BR>\n" );
document.write( "------------------------------------------------------------------------------<BR>\n" );
document.write( "D1           276.1500     14.069     19.628      0.000     247.330     304.970<BR>\n" );
document.write( "D2           277.1875     14.069     19.702      0.000     248.368     306.007<BR>\n" );
document.write( "D3           284.2125     14.069     20.201      0.000     255.393     313.032<BR>\n" );
document.write( "D4           284.7875     14.069     20.242      0.000     255.968     313.607<BR>\n" );
document.write( "==============================================================================<BR>\n" );
document.write( "Omnibus:                        5.154   Durbin-Watson:                   0.100<BR>\n" );
document.write( "Prob(Omnibus):                  0.076   Jarque-Bera (JB):                4.808<BR>\n" );
document.write( "Skew:                           0.930   Prob(JB):                       0.0904<BR>\n" );
document.write( "Kurtosis:                       2.615   Cond. No.                         1.00<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Notes:<BR>\n" );
document.write( "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\r<BR>\n" );
document.write( "\n" );
document.write( "FRIG Seasonal Indices (SI): {'D1': -132.71875000000023, 'D2': 112.65624999999977, 'D3': 214.90624999999977, 'D4': -194.84375000000023}<BR>\n" );
document.write( "DUR Seasonal Indices (SI): {'D1': -4.434375000000102, 'D2': -3.3968750000000796, 'D3': 3.6281249999999545, 'D4': 4.203124999999943}<BR>\n" );
document.write( "De-seasonalized data saved to deseasonalized_data.csv\r<BR>\n" );
document.write( "\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "[file-tag: code-generated-file-0-1760905965447315210]\r<BR>\n" );
document.write( "\n" );
document.write( "The quarterly sales data spans from 1978 to 1985 (32 observations). We use Ordinary Least Squares (OLS) regression with dummy variables to estimate the seasonal components.\r<BR>\n" );
document.write( "\n" );
document.write( "## a. Models with Intercept (Reference Quarter: Q1)\r<BR>\n" );
document.write( "\n" );
document.write( "The models estimated are:\r<BR>\n" );
document.write( "\n" );
document.write( "  * **FRIG:** $\text{FRIG} = \beta_1 + \beta_2 D_2 + \beta_3 D_3 + \beta_4 D_4 + \mu$<BR>\n" );
document.write( "  * **DUR:** $\text{DUR} = \beta_1 + \beta_2 D_2 + \beta_3 D_3 + \beta_4 D_4 + \mu$\r<BR>\n" );
document.write( "\n" );
document.write( "| Variable | $\hat{\beta}$ (FRIG) | $\hat{\beta}$ (DUR) | Interpretation ($\hat{\beta}$) |<BR>\n" );
document.write( "| :--- | :--- | :--- | :--- |<BR>\n" );
document.write( "| **const** ($\beta_1$) | $1222.13$ | $276.15$ | **Q1 Mean:** Estimated mean sales in the first quarter. |<BR>\n" );
document.write( "| **$D_2$** ($\beta_2$) | $245.38$ | $1.04$ | **Q2 Effect:** Estimated difference in sales between Q2 and Q1. |<BR>\n" );
document.write( "| **$D_3$** ($\beta_3$) | $347.63$ | $8.06$ | **Q3 Effect:** Estimated difference in sales between Q3 and Q1. |<BR>\n" );
document.write( "| **$D_4$** ($\beta_4$) | $-62.13$ | $8.64$ | **Q4 Effect:** Estimated difference in sales between Q4 and Q1. |\r<BR>\n" );
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document.write( "**Interpretation of Results:**\r<BR>\n" );
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document.write( "1.  **FRIG Sales:** The model has a significant explanatory power ($R^2=0.532$), indicating strong seasonality.\r<BR>\n" );
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document.write( "      * **Baseline (Q1 Mean):** The mean sales in the first quarter (Q1) are estimated to be $1222.13$.<BR>\n" );
document.write( "      * **Q2 and Q3 Effect:** Sales in Q2 and Q3 are significantly higher than in Q1, by $245.38$ and $347.63$ units, respectively ($P<0.01$). **Q3 has the highest estimated mean sales** ($1222.13 + 347.63 = 1569.76$).<BR>\n" );
document.write( "      * **Q4 Effect:** Sales in Q4 are slightly lower than Q1, by $62.13$ units, but this difference is **not statistically significant** ($P=0.470$).\r<BR>\n" );
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document.write( "2.  **DUR (Durable Goods) Sales:** The model has very poor explanatory power ($R^2=0.011$), indicating that seasonal dummy variables alone do not capture significant variation in Durable Goods sales.\r<BR>\n" );
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document.write( "      * **Baseline (Q1 Mean):** The mean sales in Q1 are estimated to be $276.15$.<BR>\n" );
document.write( "      * **Q2, Q3, and Q4 Effects:** The estimated differences in sales for Q2 ($1.04$), Q3 ($8.06$), and Q4 ($8.64$) relative to Q1 are all **not statistically significant** ($P>0.6$). This suggests that the seasonal pattern is either very weak or non-existent in this series, especially compared to the Frig sales.\r<BR>\n" );
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document.write( "-----\r<BR>\n" );
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document.write( "## b. Models without Intercept (All Dummies Included)\r<BR>\n" );
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document.write( "The models estimated are:\r<BR>\n" );
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document.write( "  * **FRIG:** $\text{FRIG} = \beta_1 D_1 + \beta_2 D_2 + \beta_3 D_3 + \beta_4 D_4 + \mu$<BR>\n" );
document.write( "  * **DUR:** $\text{DUR} = \beta_1 D_1 + \beta_2 D_2 + \beta_3 D_3 + \beta_4 D_4 + \mu$\r<BR>\n" );
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document.write( "| Variable | $\hat{\beta}$ (FRIG) | $\hat{\beta}$ (DUR) | Interpretation ($\hat{\beta}$) |<BR>\n" );
document.write( "| :--- | :--- | :--- | :--- |<BR>\n" );
document.write( "| **$D_1$** ($\beta_1$) | $1222.13$ | $276.15$ | **Q1 Mean:** Estimated mean sales in the first quarter. |<BR>\n" );
document.write( "| **$D_2$** ($\beta_2$) | $1467.50$ | $277.19$ | **Q2 Mean:** Estimated mean sales in the second quarter. |<BR>\n" );
document.write( "| **$D_3$** ($\beta_3$) | $1569.75$ | $284.21$ | **Q3 Mean:** Estimated mean sales in the third quarter. |<BR>\n" );
document.write( "| **$D_4$** ($\beta_4$) | $1160.00$ | $284.79$ | **Q4 Mean:** Estimated mean sales in the fourth quarter. |\r<BR>\n" );
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document.write( "**Interpretation of Results:**\r<BR>\n" );
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document.write( "This model is mathematically equivalent to the intercept model but provides the **estimated mean sales for each quarter directly** as the coefficients.\r<BR>\n" );
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document.write( "1.  **FRIG Sales:** The means clearly show the seasonal pattern: Q3 (Highest, $1569.75$) $\gg$ Q2 ($1467.50$) $\gg$ Q1 ($1222.13$) $\gg$ Q4 (Lowest, $1160.00$). All quarterly means are statistically significant ($P<0.001$).<BR>\n" );
document.write( "2.  **DUR Sales:** The quarterly means are clustered closely together (ranging from $276.15$ to $284.79$), confirming that the seasonal component is small or negligible.\r<BR>\n" );
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document.write( "-----\r<BR>\n" );
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document.write( "## c. De-seasonalize the Sales Data\r<BR>\n" );
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document.write( "### Procedure and Purpose of De-seasonalization\r<BR>\n" );
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document.write( "**Purpose:** De-seasonalization (or seasonal adjustment) is the process of removing predictable, periodic fluctuations (seasonality) from a time series. This process reveals the underlying **trend** and **cyclical** components of the data, which are crucial for forecasting, making policy decisions, and comparing data across different periods.\r<BR>\n" );
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document.write( "**Three-Step Seasonal Adjustment Procedure (Additive Model using Dummy Variables):**\r<BR>\n" );
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document.write( "1.  **Estimate the Seasonal Component ($S_i$):** The seasonal component for each quarter ($S_i$) is estimated as the mean of sales for that quarter ($\hat{\beta}_i$) from the no-intercept dummy variable model (Part b).\r<BR>\n" );
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document.write( "      * $\hat{\beta}_i$ is the estimated mean for Quarter $i$.\r<BR>\n" );
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document.write( "2.  **Calculate the Seasonal Index ($SI_i$):** The seasonal index measures the estimated seasonal effect as a deviation from the overall average sales ($\bar{Y}$).\r<BR>\n" );
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document.write( "      * $SI_i = \hat{\beta}_i - \bar{Y}$\r<BR>\n" );
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document.write( "3.  **De-seasonalize the Data ($Y_{t}^*$):** The de-seasonalized series is obtained by subtracting the seasonal index corresponding to that period from the original data point ($Y_t$).\r<BR>\n" );
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document.write( "      * $Y_{t}^* = Y_t - SI_i$\r<BR>\n" );
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document.write( "### De-seasonalization Results\r<BR>\n" );
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document.write( "The overall mean sales for each series are:\r<BR>\n" );
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document.write( "  * $\bar{\text{FRIG}} \approx 1354.84$<BR>\n" );
document.write( "  * $\bar{\text{DUR}} \approx 280.57$\r<BR>\n" );
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document.write( "The calculated Seasonal Indices ($SI$) based on the quarterly means from part (b) and the overall mean are:\r<BR>\n" );
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document.write( "| Quarter | $\hat{\beta}$ (Quarter Mean) | $SI_{\text{FRIG}}$ | $SI_{\text{DUR}}$ |<BR>\n" );
document.write( "| :--- | :--- | :--- | :--- |<BR>\n" );
document.write( "| **Q1** ($D_1$) | $1222.13$ | $-132.72$ | $-4.43$ |<BR>\n" );
document.write( "| **Q2** ($D_2$) | $1467.50$ | $112.66$ | $-3.40$ |<BR>\n" );
document.write( "| **Q3** ($D_3$) | $1569.75$ | $214.91$ | $3.63$ |<BR>\n" );
document.write( "| **Q4** ($D_4$) | $1160.00$ | $-194.84$ | $4.20$ |\r<BR>\n" );
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document.write( "The de-seasonalized data for both FRIG and DUR is provided in the file `deseasonalized_data.csv`.\r<BR>\n" );
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document.write( "**Interpretation of De-seasonalized Data:**<BR>\n" );
document.write( "The deseasonalized series represents what the sales would have been if seasonal effects had not been present. For the **FRIG** series, the deseasonalized values will be clustered much more tightly around the overall mean ($\approx 1354.84$) than the original data, making the underlying long-term **upward trend** and any **cyclical variation** clearer. Since the $SI_{\text{DUR}}$ values are very small, the deseasonalized **DUR** series is virtually identical to the original series, reinforcing the finding that seasonality is not a significant factor in durable goods sales.\r<BR>\n" );
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document.write( "The de-seasonalized data is saved to a CSV file: [File deseasonalized\_data.csv] </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );