**3. Methods**

We utilize two methods to analyze the effect of EPCs on the value of dwellings when taking the present value of the energy price into account. Both methods build on Gordon's dividend model in hedonic regression [15]. In the first method, the dividend model is utilized to calculate the expected value added of each energy label. This expected value added is compared with the estimate for the actual value added, which is estimated based on the hedonic regression. The second method is a hedonic regression model in which the energy price and the rate of discount are included by using the present value of the expected energy cost as an explanatory variable.

### *3.1. Calculating Expected Value Added*

Gordon and Shapiro's growth model is written as follows [15]:

$$PV\_0 = \frac{D\_1}{r - g} \tag{1}$$

where *PV*0 is the items value at time *t* = 0, *D*1 is the expected dividend at time *t* = 1, and *r* is the demand on return. When we use this model with respect to the dwelling and energy consumption, we can define *PV*0 as the future energy cost and *Dt* as the yearly energy cost of the dwelling. If the yearly energy cost is expected to grow with a yearly rate of *g*, we can rewrite the model as follows (The formula calculates the maximum present value (PV) of energy cost, and may overestimate the theoretical costs for dwellings that have a small remainder life expectancy. We expect most of these dwellings to fall in the G category.):

$$PV\_0 = \frac{D\_1}{r - \text{g}} = \frac{D\_0(1 + \text{g})}{r - \text{g}} \tag{2}$$

To find the yearly energy cost *D*0, we calculate the maximum energy consumption per square meter in the different energy label categories. This energy consumption is calculated based on the demands from the Norwegian Water Resources and Energy Directorate. The formula for the different labels is presented in Table 1. The only requirement for the G category is that the energy consumption is higher than in the F category. We calculate this category by assuming that energy consumption is 25% higher than in the F category, which is based on the average difference in energy consumption between the F and G categories. Table 1 shows the maximum energy consumption associated with each energy label category.

**Table 1.** Formula for calculating the maximum energy consumption associated with each energy label category [16].


For instance, a 100-square meter apartment must not exceed the maximum limit of 105 kWh/m<sup>2</sup> (95 + 10,000/100) for energy consumption to earn a grade B on its EPC. A dwelling without a heat pump and solar energy that was built in accordance with the minimum requirements of the building regulations will normally achieve a grade C. Grade B may be earned, for example, by installing a heat pump to utilize solar energy or by improving the insulation of windows. Grade A is only achieved by dwellings wherein all measures of energy efficiency are adopted. Few dwellings achieve grades A or B. Tables 2 and 3 show the distribution of energy grades for small houses and apartments, respectively. Table 4 presents the distribution after 2010 and the calculated distribution before 2010 based on our data.


**Table 2.** The distribution (number of new certificates) of energy labels for small houses [17].

**Table 3.** The distribution (number of new certificates) of energy labels for apartments [17].



**Table 4.** EPCs and sale year.

Note: The table shows the number of dwellings sold in a given year with a given EPC grade after July 2010 and the implicit EPC grade before July 2010.

Then, the energy consumption per square meter (kWh/m2) is used to calculate the yearly energy cost for each dwelling as follows:

$$Yearly\ energy\ cost = \frac{kwh}{m^2} \cdot m^2 \cdot p\_t^e \tag{3}$$

where *Pet* is the energy cost at time t measured in NOK/kwh. By applying the present value for the grade A dwellings and subtracting the present value of a dwelling in the grade B category, we find the expected value added of a grade A labeled dwelling. This value added can be compared with the value added found in the market transaction data to test whether the expected energy label premium is achieved in the market.

### *3.2. Estimating the Actual Energy Label Premium*

The real energy label price premium is estimated based on a hedonic regression model and real estate transaction data. The hedonic model is used to control for heterogeneity with respect to different characteristics, with dummy variables for the different energy labels included as follows:

$$
ln P\_n^t = \beta\_o^t + \sum\_{k=1}^K \beta\_k z\_{nk}^t + \varepsilon\_n^t \tag{4}
$$

Here, the logarithm of the dwelling price per square meter, P, is explained by a set of explanatory variables *<sup>z</sup>tnk*. The explanatory variables *z* comprise age, location, dwelling type, energy label, and dwelling size, and ε*tn* is the error term.

First, the explanatory variables are the energy labels from A to G, with F as the reference (baseline) energy label. F is chosen instead of G as the baseline because of the unique characteristics of the G category. We found that the G category includes all dwellings where sellers neglect to identify the energy label. For example, if the owner of a C label dwelling neglects to go through the labeling process, the dwelling will automatically ge<sup>t</sup> a G label. Second, the age variable measures the difference between the year of the sale and the construction year of the dwelling. As this difference probably is of less importance the older the dwelling is, we measure the age variable by 1/(sale year – construction year). This accounts for the fact that the age of a building is a relatively more important factor if we compare a brand-new dwelling with a one-year-old dwelling than if we compare a 20-year-old with a 21-year-old dwelling. Because of the way the variable is constructed, we expect it to be positively corelated with the house price. Third, we include dummy variables for location based on the different city districts in Oslo (St. Hanshaugen, Gamle Oslo, Grynerløkka og Sagene, Outer Oslo West, Outer Oslo East), where the district Frogner is used as the baseline (it would have been preferable to include smaller, and hence more urban, districts, but the number was chosen based on the number of observations. See Marmolejo-Duarte and Chen (2019) [18]). Fourth, we control for dwelling type, where we separate single-family houses, townhouses, and semidetached houses with dummies, and where apartment is the baseline category. Fifth, we also include dummy variables for different size of the dwellings. Small is a dummy for dwellings between 50–80 m2, medium is dummy for dwellings from 81–120 m2, and large is dummy for dwellings >120 m2. Hence, the baseline size is <50 m2. We use the log-linear (semilog) functional form in the regressions because it makes it easier to interpret the coefficients and because the semilog functional form is known to mitigate the problem of heteroscedasticity [19]. In total, we observe (T + 1) periods. Note that if we ignore the year dummies and the time subscript, we are left with a standard hedonic model. Based on Equation (4), we are hence able to estimate two models, the post-label hedonic model, Model 1, and the pre-label hedonic model, Model 2.

Based on the results from the model, we estimate the price per square meter for the reference dwelling, which is a dwelling <50 m<sup>2</sup> located in Frogner. We set the age of this dwelling at five years and calculate the square meter price for all energy label categories. The price difference between the different labels is the actual value added achieved in the market compared with the expected value added.

### *3.3. Hedonic Model With Energy Price and Rate of Discount*

To examine how the energy label, energy price, and rate of discount affect the price of the dwelling, a hedonic model is constructed where these three factors are represented through the expected energy cost. Hence, we apply the same model as described above, but replace the energy label dummies with the expected value of the energy costs given by:

$$LN(PV\,\text{energy cost}) = LN\left(\frac{k\mathcal{W}h \cdot P\_t^\ast}{r\_t - g\_t}\right) \tag{5}$$

Here, the logarithm of the present value per square meter, PV, is given by taking the logarithm of energy consumption per square meter times the expected energy cost per square meter, divided by the discount rate minus the growth rate. Figure 1 illustrates the development in energy price and discount rate over the time period 2000–2014.

**Figure 1.** Energy price and interest rate year 2000–2014. The interest rate is given by the Norwegian 10-year governmen<sup>t</sup> bond and the energy price is from the energy price area of Oslo [20–22].
